235 solved out of 808 shown (show only solved or open)
OPEN - $500 If$A\subseteq \{1,\ldots,N\}$with$\lvert A\rvert=n$is such that the subset sums$\sum_{a\in S}a$are distinct for all$S\subseteq A$then $N \gg 2^{n}.$ Erdős called this 'perhaps my first serious problem'. The powers of$2$show that$2^n$would be best possible here. The trivial lower bound is$N \gg 2^{n}/n$, since all$2^n$distinct subset sums must lie in$[0,Nn)$. Erdős and Moser [Er56] proved $N\geq (\tfrac{1}{4}-o(1))\frac{2^n}{\sqrt{n}}.$ A number of improvements of the constant have been given (see [St23] for a history), with the current record$\sqrt{2/\pi}$first proved in unpublished work of Elkies and Gleason. Two proofs achieving this constant are provided by Dubroff, Fox, and Xu [DFX21], who in fact prove the exact bound$N\geq \binom{n}{\lfloor n/2\rfloor}$. In [Er73] and [ErGr80] the generalisation where$A\subseteq (0,N]$is a set of real numbers such that the subset sums all differ by at least$1$is proposed, with the same conjectured bound. (The second proof of [DFX21] applies also to this generalisation.) This problem appears in Erdős' book with Spencer [ErSp74] in the final chapter titled 'The kitchen sink'. As Ruzsa writes in [Ru99] "it is a rich kitchen where such things go to the sink". See also [350]. Additional thanks to: Zachary Hunter SOLVED -$1000
Can the smallest modulus of a covering system be arbitrarily large?
Described by Erdős as 'perhaps my favourite problem'. Hough [Ho15], building on work of Filaseta, Ford, Konyagin, Pomerance, and Yu [FFKPY07], has shown the answer is no: the smallest modulus must be at most $10^{18}$.

An alternative, simpler, proof was given by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22], who improved the bound on the smallest modulus to $616000$.

OPEN - $5000 If$A\subseteq \mathbb{N}$has$\sum_{n\in A}\frac{1}{n}=\infty$then must$A$contain arbitrarily long arithmetic progressions? This is essentially asking for good bounds on$r_k(N)$, the size of the largest subset of$\{1,\ldots,N\}$without a non-trivial$k$-term arithmetic progression. For example, a bound like $r_k(N) \ll_k \frac{N}{(\log N)(\log\log N)^2}$ would be sufficient. Even the case$k=3$is non-trivial, but was proved by Bloom and Sisask [BlSi20]. Much better bounds for$r_3(N)$were subsequently proved by Kelley and Meka [KeMe23]. Green and Tao [GrTa17] proved$r_4(N)\ll N/(\log N)^{c}$for some small constant$c>0$. Gowers [Go01] proved $r_k(N) \ll \frac{N}{(\log\log N)^{c_k}},$ where$c_k>0$is a small constant depending on$k$. The current best bounds for general$k$are due to Leng, Sah, and Sawhney [LSS24], who show that $r_k(N) \ll \frac{N}{\exp((\log\log N)^{c_k})}$ for some constant$c_k>0$depending on$k$. Curiously, Erdős [Er83c] thought this conjecture was the 'only way to approach' the conjecture that there are arbitrarily long arithmetic progressions of prime numbers, now a theorem due to Green and Tao [GrTa08]. In [Er81] Erdős makes the stronger conjecture that $r_k(N) \ll_C\frac{N}{(\log N)^C}$ for every$C>0$(now known for$k=3$due to Kelley and Meka [KeMe23]) - see [140]. See also [139] and [142]. SOLVED -$10000
For any $C>0$ are there infinitely many $n$ such that $p_{n+1}-p_n> C\frac{\log\log n\log\log\log\log n}{(\log\log \log n)^2}\log n?$
The peculiar quantitative form of Erdős' question was motivated by an old result of Rankin [Ra38], who proved there exists some constant $C>0$ such that the claim holds. Solved by Maynard [Ma16] and Ford, Green, Konyagin, and Tao [FGKT16]. The best bound available, due to all five authors [FGKMT18], is that there are infinitely many $n$ such that $p_{n+1}-p_n\gg \frac{\log\log n\log\log\log\log n}{\log\log \log n}\log n.$ The likely truth is a lower bound like $\gg(\log n)^2$. In [Er97c] Erdős revised the value of this problem to \$5000 and reserved the \$10000 for a lower bound of $>(\log n)^{1+c}$ for some $c>0$.

OPEN
Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that $\lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C?$
Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\log n$. This problem asks whether $S=[0,\infty]$. Although this conjecture remains unproven, a lot is known about $S$. Some highlights:
• $\infty\in S$ by Westzynthius' result [We31] on large prime gaps,
• $0\in S$ by the work of Goldston, Pintz, and Yildirim [GPY09] on small prime gaps,
• Erdős [Er55] and Ricci [Ri56] independently showed that $S$ has positive Lebesgue measure,
• Hildebrand and Maier [HiMa88] showed that $S$ contains arbitrarily large (finite) numbers,
• Pintz [Pi16] showed that there exists some small constant $c>0$ such that $[0,c]\subset S$,
• Banks, Freiberg, and Maynard [BFM16] showed that at least $12.5\%$ of $[0,\infty)$ belongs to $S$,
• Merikoski [Me20] showed that at least $1/3$ of $[0,\infty)$ belongs to $S$, and that $S$ has bounded gaps.
In [Er97c] Erdős asks whether $S$ is everywhere dense.
SOLVED - $100 Let$d_n=p_{n+1}-p_n$. Are there infinitely many$n$such that$d_n<d_{n+1}<d_{n+2}$? Conjectured by Erdős and Turán [ErTu48]. Shockingly Erdős offered \$25000 for a disproof of this, but as he comments, it 'is certainly true'.

Indeed, the answer is yes, as proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the Maynard-Tao machinery concerning bounded gaps between primes [Ma15]. They in fact prove that, for any $m\geq 1$, there are infinitely many $n$ such that $d_n<d_{n+1}<\cdots <d_{n+m}$ and infinitely many $n$ such that $d_n> d_{n+1}>\cdots >d_{n+m}.$

OPEN
Is there a covering system all of whose moduli are odd?
Asked by Erdős and Selfridge (sometimes also with Schinzel). They also asked whether there can be a covering system such that all the moduli are odd and squarefree. The answer to this stronger question is no, proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22].

Hough and Nielsen [HoNi19] proved that at least one modulus must be divisible by either $2$ or $3$. A simpler proof of this fact was provided by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22].

Selfridge has shown (as reported in [Sc67]) that such a covering system exists if a covering system exists with moduli $n_1,\ldots,n_k$ such that no $n_i$ divides any other $n_j$ (but the latter has been shown not to exist, see [586]).

SOLVED
For any finite colouring of the integers is there a covering system all of whose moduli are monochromatic?
Conjectured by Erdős and Graham, who also ask about a density-type version: for example, is $\sum_{\substack{a\in A\\ a>N}}\frac{1}{a}\gg \log N$ a sufficient condition for $A$ to contain the moduli of a covering system? The answer (to both colouring and density versions) is no, due to the result of Hough [Ho15] on the minimum size of a modulus in a covering system - in particular one could colour all integers $<10^{18}$ different colours and all other integers a new colour.
OPEN
Let $A$ be the set of all integers not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?
Crocker [Cr71] has proved there are are $\gg\log\log N$ such integers in $\{1,\ldots,N\}$ (any number of the form $2^{2^m}-1$ with $m\geq 3$ suffices). Pan [Pa11] improved this to $\gg_\epsilon N^{1-\epsilon}$ for any $\epsilon>0$. Erdős believes this cannot be proved by covering systems, i.e. integers of the form $p+2^k+2^l$ exist in every infinite arithmetic progression.

OPEN
Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of 2?
Erdős described this as 'probably unattackable'. In [ErGr80] Erdős and Graham suggest that no such $k$ exists. Gallagher [Ga75] has shown that for any $\epsilon>0$ there exists $k(\epsilon)$ such that the set of integers which are the sum of a prime and at most $k(\epsilon)$ many powers of 2 has lower density at least $1-\epsilon$.

Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of 2 suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. (The restriction to odd integers is important here - for example, Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ (see [9]) suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).

OPEN
Is every odd $n$ the sum of a squarefree number and a power of 2?
Odlyzko has checked this up to $10^7$. Granville and Soundararajan [GrSo98] have proved that this is very related to the problem of finding primes $p$ for which $2^p\equiv 2\pmod{p^2}$ (for example this conjecture implies there are infinitely many such $p$).

This is equivalent to asking whether every $n$ not divisible by $4$ is the sum of a squarefree number and a power of two. Erdős thought that proving this with two powers of 2 is perhaps easy, and could prove that it is true (with a single power of two) for almost all $n$.

OPEN
Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with $\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>0?$ Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with $\lvert A\cap\{1,\ldots,N\}\rvert<N^{1-c}?$

Is it true that $\sum_{n\in A}\frac{1}{n}<\infty?$

Asked by Erdős and Sárközy [ErSa70], who proved that $A$ must have density $0$. They also prove that this is essentially best possible, in that given any function $f(x)\to \infty$ as $x\to \infty$ there exists a set $A$ with this property and infinitely many $N$ such that $\lvert A\cap\{1,\ldots,N\}\rvert>\frac{N}{f(N)}.$ (Their example is given by all integers in $(y_i,\frac{3}{2}y_i)$ congruent to $1$ modulo $(2y_{i-1})!$, where $y_i$ is some sufficiently quickly growing sequence.)

An example of an $A$ with this property where $\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\log N>0$ is given by the set of $p^2$, where $p\equiv 3\pmod{4}$ is prime.

For the finite version see [13].

SOLVED - $100 Let$A\subseteq \{1,\ldots,N\}$be such that there are no$a,b,c\in A$such that$a\mid(b+c)$and$a<\min(b,c)$. Is it true that$\lvert A\rvert\leq N/3+O(1)$? Asked by Erdős and Sárközy, who observed that$(2N/3,N]\cap \mathbb{N}$is such a set. The answer is yes, as proved by Bedert [Be23]. For the infinite version see [12]. Additional thanks to: Zachary Chase OPEN Let$A\subseteq \mathbb{N}$. Let$B\subseteq \mathbb{N}$be the set of integers which are representable in exactly one way as the sum of two elements from$A$. Is it true that for all$\epsilon>0$and large$N$$\lvert \{1,\ldots,N\}\backslash B\rvert \gg_\epsilon N^{1/2-\epsilon}.$ Asked by Erdős, Sárközy, and Szemerédi, who constructed an$A$such that for all$\epsilon>0$and all large$N$$\lvert \{1,\ldots,N\}\backslash B\rvert \ll_\epsilon N^{1/2+\epsilon},$ and yet there for all$\epsilon>0$there exist infinitely many$N$where $\lvert \{1,\ldots,N\}\backslash B\rvert \gg_\epsilon N^{1/3-\epsilon}.$ Erdös and Freud investigated the finite analogue in 'a recent Hungarian paper', proving that there exists$A\subseteq \{1,\ldots,N\}$such that the number of integers not representable in exactly one way as the sum of two elements from$A$is$<2^{3/2}N^{1/2}$, and suggest the constant$2^{3/2}$is perhaps best possible. OPEN Is it true that $\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}$ converges, where$p_n$is the sequence of primes? Erdős suggested that a computer could be used to explore this, and did not see any other method to attack this. Tao [Ta23] has proved that this series does converge assuming a strong form of the Hardy-Littlewood prime tuples conjecture. SOLVED Is the set of odd integers not of the form$2^k+p$the union of an infinite arithmetic progression and a set of density$0$? Erdős called this conjecture 'rather silly'. Using covering congruences Erdős [Er50] proved that the set of such odd integers contains an infinite arithmetic progression. Chen [Ch23] has proved the answer is no. See also [9], [10], and [11]. OPEN Are there infinitely many primes$p$such that every even number$n\leq p-3$can be written as a difference of primes$n=q_1-q_2$where$q_1,q_2\leq p$? The first prime without this property is$97$. OPEN We call$m$practical if every integer$n<m$is the sum of distinct divisors of$m$. If$m$is practical then let$h(m)$be such that$h(m)$many divisors always suffice. Are there infinitely many practical$m$such that $h(m) < (\log\log m)^{O(1)}?$ Is it true that$h(n!)<n^{o(1)}$? Or perhaps even$h(n!)<(\log n)^{O(1)}$? It is easy to see that almost all numbers are not practical. Erdős originally showed that$h(n!) <n$. Vose [Vo85] proved the existence of infinitely many practical$m$such that$h(m)\ll (\log m)^{1/2}$. Related to [304]. SOLVED -$500
If $G$ is an edge-disjoint union of $n$ copies of $K_n$ then is $\chi(G)=n$?
Conjectured by Faber, Lovász, and Erdős (apparently 'at a party in Boulder, Colarado in September 1972' [Er81]).

Kahn [Ka92] proved that $\chi(G)\leq (1+o(1))n$ (for which Erdős gave him a 'consolation prize' of \$100). Hindman has proved the conjecture for$n<10$. Kang, Kelly, Kühn, Methuku, and Osthus [KKKMO21] have proved the answer is yes for all sufficiently large$n$. In [Er97d] Erdős asks how large$\chi(G)$can be if instead of asking for the copies of$K_n$to be edge disjoint we only ask for their intersections to be triangle free, or to contain at most one edge. OPEN -$1000
Let $f(n,k)$ be minimal such that every $\mathcal{F}$ family of $n$-uniform sets with $\lvert F\rvert \geq f(n,k)$ contains a $k$-sunflower. Is it true that $f(n,k) < c_k^n$ for some constant $c_k>0$?
Erdős and Rado [ErRa60] originally proved $f(n,k)\leq (k-1)^nn!$. Kostochka [Ko97] improved this slightly (in particular establishing an upper bound of $o(n!)$, for which Erdős awarded him the consolation prize of \$100), but the bound stood at$n^{(1+o(1))n}$for a long time until Alweiss, Lovett, Wu, and Zhang [ALWZ20] proved $f(n,k) < (Ck\log n\log\log n)^n$ for some constant$C>1$. This was refined slightly, independently by Rao [Ra20], Frankston, Kahn, Narayanan, and Park [FKNP19], and Bell, Chueluecha, and Warnke [BCW21], leading to the current record of $f(n,k) < (Ck\log n)^n$ for some constant$C>1$. In [Er81] offered \$1000 for a proof or disproof even just in the special case when $k=3$, which he expected 'contains the whole difficulty'. He also wrote 'I really do not see why this question is so difficult'.

The usual focus is on the regime where $k=O(1)$ is fixed (say $k=3$) and $n$ is large, although for the opposite regime Kostochka, Rödl, and Talysheva [KoRoTa99] have shown $f(n,k)=(1+O_n(k^{-1/2^n}))k^n.$

SOLVED - $500 Let$f(n)$be minimal such that there is an intersecting family$\mathcal{F}$of sets of size$n$(so$A\cap B\neq\emptyset$for all$A,B\in \mathcal{F}$) with$\lvert \mathcal{F}\rvert=f(n)$such that any set$S$with$\lvert S\rvert \leq n-1$is disjoint from at least one$A\in \mathcal{F}$. Is it true that $f(n) \ll n?$ Conjectured by Erdős and Lovász [ErLo75], who proved that $\frac{8}{3}n-3\leq f(n) \ll n^{3/2}\log n$ for all$n$. The upper bound was improved by Kahn [Ka92b] to $f(n) \ll n\log n.$ (The upper bound constructions in both cases are formed by taking a random set of lines from a projective plane of order$n-1$, assuming$n-1$is a prime power.) This problem was solved by Kahn [Ka94] who proved the upper bound$f(n) \ll n$. The Erdős-Lovász lower bound of$\frac{8}{3}n-O(1)$has not been improved, and it has been speculated (see e.g. [Ka94]) that the correct answer is$3n+O(1)$. In [Er97f] Erdős asks about$f_\epsilon(n)$, defined analogously except with$\lvert S\rvert \leq n-1$replaced by$\lvert S\rvert \leq (1-\epsilon)n$. He asks whether$f_\epsilon(n)/n\to \infty$as$\epsilon \to 0$. Additional thanks to: Zachary Chase SOLVED Let$\epsilon>0$and$n$be sufficiently large depending on$\epsilon$. Is there a graph on$n$vertices with$\geq n^2/8$many edges which contains no$K_4$such that the largest independent set has size at most$\epsilon n$? In other words, if$\mathrm{rt}(n;k,\ell)$is the Ramsey-Turán number then is it true that (for sufficiently large$n$) $\mathrm{rt}(n; 4,\epsilon n)\geq n^2/8?$ Conjectured by Bollobás and Erdős [BoEr76], who proved the existence of such a graph with$(1/8+o(1))n^2$many edges. Solved by Fox, Loh, and Zhao [FLZ15], who proved that for every$n\geq 1$there exists a graph on$n$vertices with$\geq n^2/8$many edges, containing no$K_4$, whose largest independent set has size at most $\ll \frac{(\log\log n)^{3/2}}{(\log n)^{1/2}}n.$ See also [615]. Additional thanks to: Mehtaab Sawhney OPEN Can every triangle-free graph on$5n$vertices be made bipartite by deleting at most$n^2$edges? The blow-up of$C_5$shows that this would be the best possible. The best known bound is due to Balogh, Clemen, and Lidicky [BCL21], who proved that deleting at most$1.064n^2$edges suffices. Additional thanks to: Casey Tompkins SOLVED Does every triangle-free graph on$5n$vertices contain at most$n^5$copies of$C_5$? Győri proved this with$1.03n^5$, which has been improved by Füredi. The answer is yes, as proved independently by Grzesik [Gr12] and Hatami, Hladky, Král, Norine, and Razborov [HHKNR13]. In [Er97f] Erdős asks more generally: if$r\geq 5$is odd and a graph has$rn$vertices and the smallest odd cycle has size$r$then is the number of cycles of size$r$at most$n^{r}$? Additional thanks to: Casey Tompkins, Tuan Tran OPEN Let$n_1<n_2<\cdots$be an arbitrary sequence of integers, each with an associated residue class$a_i\pmod{n_i}$. Let$A$be the set of integers$n$such that for every$i$either$n<n_i$or$n\not\equiv a_i\pmod{n_i}$. Must the logarithmic density of$A$exist? SOLVED Let$A\subset\mathbb{N}$be infinite. Must there exist some$k\geq 1$such that almost all integers have a divisor of the form$a+k$for some$a\in A$? Asked by Erdős and Tenenbaum. Ruzsa gave the following simple counterexample: let$A=\{n_1<n_2<\cdots \}$where$n_l \equiv -(k-1)\pmod{p_k}$for all$k\leq l$, where$p_k$denotes the$k$th prime. Tenenbaum asked the weaker variant (still open) where for every$\epsilon>0$there is some$k=k(\epsilon)$such that at least$1-\epsilon$density of all integers have a divisor of the form$a+k$for some$a\in A$. Additional thanks to: Imre Ruzsa SOLVED -$100
An $\epsilon$-almost covering system is a set of congruences $a_i\pmod{n_i}$ for distinct moduli $n_1<\ldots<n_k$ such that the density of those integers which satisfy none of them is $\leq \epsilon$. Is there a constant $C>1$ such that for every $\epsilon>0$ and $N\geq 1$ there is an $\epsilon$-almost covering system with $N\leq n_1<\cdots <n_k\leq CN$?
By a simple averaging argument the set of moduli $[m_1,m_2]\cap \mathbb{N}$ has a choice of residue classes which form an $\epsilon(m_1,m_2)$-almost covering system with $\epsilon(m_1,m_2)=\prod_{m_1\leq m\leq m_2}(1-1/m).$ A $0$-covering system is just a covering system, and so by Hough [Ho15] these only exist for $n_1<10^{18}$.

The answer is no, as proved by Filaseta, Ford, Konyagin, Pomerance, and Yu [FFKPY07], who (among other results) prove that if $1< C \leq N^{\frac{\log\log\log N}{4\log\log N}}$ then, for any $N\leq n_1<\cdots< n_k\leq CN$, the density of integers not covered for any fixed choice of residue classes is at least $\prod_{i}(1-1/n_i)$ (and this density is achieved for some choice of residue classes as above).

OPEN - $500 If$A\subseteq \mathbb{N}$is such that$A+A$contains all but finitely many integers then$\limsup 1_A\ast 1_A(n)=\infty$. Conjectured by Erdős and Turán. They also suggest the stronger conjecture that$\limsup 1_A\ast 1_A(n)/\log n>0$. Another stronger conjecture would be that the hypothesis$\lvert A\cap [1,N]\rvert \gg N^{1/2}$for all large$N$suffices. Erdős and Sárközy conjectured the stronger version that if$A=\{a_1<a_2<\cdots\}$and$B=\{b_1<b_2<\cdots\}$with$a_n/b_n\to 1$are such that$A+B=\mathbb{N}$then$\limsup 1_A\ast 1_B(n)=\infty$. See also [40]. SOLVED -$100
Is there an explicit construction of a set $A\subseteq \mathbb{N}$ such that $A+A=\mathbb{N}$ but $1_A\ast 1_A(n)=o(n^\epsilon)$ for every $\epsilon>0$?
The existence of such a set was asked by Sidon to Erdős in 1932. Erdős (eventually) proved the existence of such a set using probabilistic methods. This problem asks for a constructive solution.

An explicit construction was given by Jain, Pham, Sawhney, and Zakharov [JPSZ24].

OPEN - $1000 Let$h(N)$be the maximum size of a Sidon set in$\{1,\ldots,N\}$. Is it true that, for every$\epsilon>0$, $h(N) = N^{1/2}+O_\epsilon(N^\epsilon)?$ A problem of Erdős and Turán. It may even be true that$h(N)=N^{1/2}+O(1)$, but Erdős remarks this is perhaps too optimistic. Erdős and Turán [ErTu41] proved an upper bound of$N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindström [Li69]. Both proofs in fact give $h(N) \leq N^{1/2}+N^{1/4}+1.$ Balogh, Füredi, and Roy [BFR21] improved the bound in the error term to$0.998N^{1/4}$, which has been further optimised by O'Bryant [OB22] to yield $h(N)\leq N^{1/2}+0.99703N^{1/4}$ for sufficiently large$N$. See also [241]. Additional thanks to: Zachary Hunter SOLVED Given any infinite set$A\subset \mathbb{N}$there is a set$B$of density$0$such that$A+B$contains all except finitely many integers. Conjectured by Erdős and Straus. Proved by Lorentz [Lo54]. OPEN Is there a set$A\subset\mathbb{N}$such that $\lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2)$ and such that every large integer can be written as$p+a$for some prime$p$and$a\in A$? Can the bound$O(\log N)$be achieved? Must such an$A$satisfy $\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{\log N}> 1?$ Such a set is called an additive complement to the primes. Erdős [Er54] proved that such a set$A$exists with$\lvert A\cap\{1,\ldots,N\}\rvert\ll (\log N)^2$(improving a previous result of Lorentz [Lo54] who achieved$\ll (\log N)^3$). Wolke [Wo96] has shown that such a bound is almost true, in that we can achieve$\ll (\log N)^{1+o(1)}$if we only ask for almost all integers to be representable. The answer to the third question is yes: Ruzsa [Ru98c] has shown that we must have $\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{\log N}\geq e^\gamma\approx 1.781.$ OPEN Let$A\subset\mathbb{N}$be such that every large integer can be written as$n^2+a$for some$a\in A$and$n\geq 0$. What is the smallest possible value of $\limsup \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}?$ Erdős observed that this value is finite and$>1$. Additional thanks to: Timothy Gowers SOLVED For any permutation$\pi\in S_n$of$\{1,\ldots,n\}$let$S(\pi)$count the number of distinct consecutive sums, that is, sums of the shape$\sum_{u\leq i\leq v}\pi(i)$. Is it true that $S(\pi) = o(n^2)$ for all$\pi\in S_n$? It is easy to see that$S(\iota)=o(n^2)$if$\iota$denotes the identity permutation, as studied by Erdős and Harzheim [Er77]. Motivated by this, Erdős asked if this remains true for all permutations. This is extremely false, as shown by Konieczny [Ko15], who both constructs an explicit permutation with$S(\pi) \geq n^2/4$, and also shows that for a random permutation we have $S(\pi)\sim \frac{1+e^{-2}}{4}n^2.$ See also [356] and [357]. SOLVED Let$B\subseteq\mathbb{N}$be an additive basis of order$k$with$0\in B$. Is it true that for every$A\subseteq\mathbb{N}$we have $d_s(A+B)\geq \alpha+\frac{\alpha(1-\alpha)}{k},$ where$\alpha=d_s(A)$and $d_s(A) = \inf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N}$ is the Schnirelmann density? Erdős [Er36c] proved this is true with$k$replaced by$2k$in the denominator (in a stronger form that only considers$A\cup (A+b)$for some$b\in B$, see [38]). Ruzsa has observed that this follows immediately from the stronger fact proved by Plünnecke [Pl70] that (under the same assumptions) $d_S(A+B)\geq \alpha^{1-1/k}.$ Additional thanks to: Imre Ruzsa OPEN Find the optimal constant$c>0$such that the following holds. For all sufficiently large$N$, if$A\sqcup B=\{1,\ldots,2N\}$is a partition into two equal parts, so that$\lvert A\rvert=\lvert B\rvert=N$, then there is some$x$such that the number of solutions to$a-b=x$with$a\in A$and$b\in B$is at least$cN$. The minimum overlap problem. The example (with$N$even)$A=\{N/2+1,\ldots,3N/2\}$shows that$c\leq 1/2$(indeed, Erdős initially conjectured that$c=1/2$). The lower bound of$c\geq 1/4$is trivial, and Scherk improved this to$1-1/\sqrt{2}=0.29\cdots$. The current records are $0.379005 < c < 0.380926\cdots,$ the lower bound due to White [Wh22] and the upper bound due to Haugland [Ha16]. SOLVED We say that$A\subset \mathbb{N}$is an essential component if$d_s(A+B)>d_s(B)$for every$B\subset \mathbb{N}$with$0<d_s(B)<1$where$d_s$is the Schnirelmann density. Can a lacunary set$A\subset\mathbb{N}$be an essential component? The answer is no by Ruzsa [Ru87], who proved that if$A$is an essential component then there exists some constant$c>0$such that$\lvert A\cap \{1,\ldots,N\}\rvert \geq (\log N)^{1+c}$for all large$N$. OPEN Does there exist$B\subset\mathbb{N}$which is not an additive basis, but is such that for every set$A\subseteq\mathbb{N}$of Schnirelmann density$\alpha$and every$N$there exists$b\in B$such that $\lvert (A\cup (A+b))\cap \{1,\ldots,N\}\rvert\geq (\alpha+f(\alpha))N$ where$f(\alpha)>0$for$0<\alpha <1 $? The Schnirelmann density is defined by $d_s(A) = \inf_{N\geq 1}\frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N}.$ Erdős [Er36c] proved that if$B$is an additive basis of order$k$then, for any set$A$of Schnirelmann density$\alpha$, for every$N$there exists some integer$b\in B$such that $\lvert (A\cup (A+b))\cap \{1,\ldots,N\}\rvert\geq \left(\alpha+\frac{\alpha(1-\alpha)}{2k}\right)N.$ It seems an interesting question (not one that Erdős appears to have asked directly, although see [35]) to improve the lower bound here, even in the case$B=\mathbb{N}$. Erdős observed that a random set of density$\alpha$shows that the factor of$\frac{\alpha(1-\alpha)}{2}$in this case cannot be improved past$\alpha(1-\alpha)$. This is a stronger propery than$B$being an essential component (see [37]). Linnik [Li42] gave the first construction of an essential component which is not an additive basis. Additional thanks to: Terence Tao OPEN -$500
Is there an infinite Sidon set $A\subset \mathbb{N}$ such that $\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$ for all $\epsilon>0$?
The trivial greedy construction achieves $\gg N^{1/3}$. The current best bound of $\gg N^{\sqrt{2}-1+o(1)}$ is due to Ruzsa [Ru98]. (Erdős [Er73] had offered \$25 for any construction which achieves$N^{c}$for some$c>1/3$.) Erdős proved that for every infinite Sidon set$A$we have $\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0,$ and also that there is a set$A\subset \mathbb{N}$with$\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$such that$1_A\ast 1_A(n)=O(1)$. Erdős and Rényi have constructed, for any$\epsilon>0$, a set$A$such that $\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$ for all large$N$and$1_A\ast 1_A(n)\ll_\epsilon 1$for all$n$. OPEN -$500
For what functions $g(N)\to \infty$ is it true that $\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}g(N)$ implies $\limsup 1_A\ast 1_A(n)=\infty$?
It is possible that this is true even with $g(N)=O(1)$, from which the Erdős-Turán conjecture [28] would follow.
OPEN - $500 Let$A\subset\mathbb{N}$be an infinite set such that the triple sums$a+b+c$are all distinct for$a,b,c\in A$(aside from the trivial coincidences). Is it true that $\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0?$ Erdős proved that if the pairwise sums$a+b$are all distinct aside from the trivial coincidences then $\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0.$ Additional thanks to: Zachary Chase OPEN Let$M\geq 1$and$N$be sufficiently large in terms of$M$. Is it true that for every maximal Sidon set$A\subset \{1,\ldots,N\}$there is another Sidon set$B\subset \{1,\ldots,N\}$of size$M$such that$(A-A)\cap(B-B)=\{0\}$? OPEN -$100
If $A,B\subset \{1,\ldots,N\}$ are two Sidon sets such that $(A-A)\cap(B-B)=\{0\}$ then is it true that $\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+O(1),$ where $f(N)$ is the maximum possible size of a Sidon set in $\{1,\ldots,N\}$? If $\lvert A\rvert=\lvert B\rvert$ then can this bound be improved to $\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq (1-c)\binom{f(N)}{2}$ for some constant $c>0$?
OPEN
Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M=M(\epsilon)$ and $B\subset \{N+1,\ldots,M\}$ such that $A\cup B\subset \{1,\ldots,M\}$ is a Sidon set of size at least $(1-\epsilon)M^{1/2}$?
SOLVED
Let $k\geq 2$. Is there an integer $n_k$ such that, if $D=\{ 1<d<n_k : d\mid n_k\}$, then for any $k$-colouring of $D$ there is a monochromatic subset $D'\subseteq D$ such that $\sum_{d\in D'}\frac{1}{d}=1$?
This follows from the colouring result of Croot [Cr03]. Croot's result allows for $n_k \leq e^{C^k}$ for some constant $C>1$ (simply taking $n_k$ to be the lowest common multiple of some interval $[1,C^k]$). Sawhney has observed that there is also a doubly exponential lower bound, and hence this bound is essentially sharp.

Indeed, we must trivially have $\sum_{d|n_k}1/d \geq k$, or else there is a greedy colouring as a counterexample. Since $\prod_{p}(1+1/p^2)$ is finite we must have $\prod_{p|n_k}(1+1/p)\gg k$. To achieve the minimal $\prod_{p|n_k}p$ we take the product of primes up to $T$ where $\prod_{p\leq T}(1+1/p)\gg k$; by Mertens theorems this implies $T\geq C^{k}$ for some constant $C>1$, and hence $n_k\geq \prod_{p\mid n_k}p\geq \exp(cC^k)$ for some $c>0$.

SOLVED
Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$?
The answer is yes, as proved by Croot [Cr03].
SOLVED - $100 If$\delta>0$and$N$is sufficiently large in terms of$\delta$, and$A\subseteq\{1,\ldots,N\}$is such that$\sum_{a\in A}\frac{1}{a}>\delta \log N$then must there exist$S\subseteq A$such that$\sum_{n\in S}\frac{1}{n}=1$? Solved by Bloom [Bl21], who showed that the quantitative threshold $\sum_{n\in A}\frac{1}{n}\gg \frac{\log\log\log N}{\log\log N}\log N$ is sufficient. This was improved by Liu and Sawhney [LiSa24] to $\sum_{n\in A}\frac{1}{n}\gg (\log N)^{4/5+o(1)}.$ Erdős speculated that perhaps even$\gg (\log\log N)^2$might be sufficient. (A construction of Pomerance, as discussed in the appendix of [Bl21], shows that this would be best possible.) SOLVED Are there infinitely many integers$n,m$such that$\phi(n)=\sigma(m)$? This would follow immediately from the twin prime conjecture. The answer is yes, proved by Ford, Luca, and Pomerance [FLP10]. SOLVED Let$A=\{a_1<\cdots<a_t\}\subseteq \{1,\ldots,N\}$be such that$\phi(a_1)<\cdots<\phi(a_t)$. The primes are such an example. Are they the largest possible? Can one show that$\lvert A\rvert<(1+o(1))\pi(N)$or even$\lvert A\rvert=o(N)$? Erdős remarks that the last conjecture is probably easy, and that similar questions can be asked about$\sigma(n)$. Solved by Tao [Ta23b], who proved that $\lvert A\rvert \leq \left(1+O\left(\frac{(\log\log x)^5}{\log x}\right)\right)\pi(x).$ In [Er95c] Erdős further asks about the situation when$\phi(a_1)\leq \cdots \leq \phi(a_t)$. OPEN -$250
Schoenberg proved that for every $c\in [0,1]$ the density of $\{ n\in \mathbb{N} : \phi(n)<cn\}$ exists. Let this density be denoted by $f(c)$. Is it true that there are no $x$ such that $f'(x)$ exists and is positive?
OPEN
Is there an infinite set $A\subset \mathbb{N}$ such that for every $a\in A$ there is an integer $n$ such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer then $n_a/a\to \infty$ as $a\to\infty$?
Carmichael has asked whether there is an integer $t$ for which $\phi(n)=t$ has exactly one solution. Erdős has proved that if such a $t$ exists then there must be infinitely many such $t$.

OPEN - $250 Let$A$be a finite set of integers. Is it true that for every$\epsilon>0$$\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?$ The sum-product problem. Erdős and Szemerédi [ErSz83] proved a lower bound of$\lvert A\rvert^{1+c}$for some constant$c>0$, and an upper bound of$o(\lvert A\rvert^2)$. The lower bound has been improved a number of times. The current record is $\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{1558}{1167}-o(1)}$ due to Rudnev and Stevens [RuSt22] (note$1558/1167=1.33504\cdots$). There is likely nothing special about the integers in this question, and indeed Erdős and Szemerédi also ask a similar question about finite sets of real or complex numbers. The current best bound for sets of reals is the same bound of Rudnev and Stevens above. The best bound for complex numbers is $\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{5}{4}},$ due to Solymosi [So05]. One can in general ask this question in any setting where addition and multiplication are defined (once one avoids any trivial obstructions such as zero divisors or finite subfields). For example, it makes sense for subsets of finite fields. The current record is that if$A\subseteq \mathbb{F}_p$with$\lvert A\rvert <p^{5/8}$then $\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{11}{9}+o(1)},$ due to Rudnev, Shakan, and Shkredov [RSS20]. There is also a natural generalisation to higher-fold sum and product sets. For example, in [ErSz83] Erdős and Szemerédi also conjecture that for any$m\geq 2$and finite set of integers$A$$\max( \lvert mA\rvert,\lvert A^m\rvert)\gg \lvert A\rvert^{m-o(1)}.$ See [53] for more on this generalisation and [808] for a stronger form of the original conjecture. Additional thanks to: Akshat Mudgal SOLVED Let$A$be a finite set of integers. Is it true that, for every$k$, if$\lvert A\rvert$is sufficiently large depending on$k$, then there are least$\lvert A\rvert^k$many integers which are either the sum or product of distinct elements of$A$? Asked by Erdős and Szemerédi [ErSz83]. Solved in this form by Chang [Ch03]. See also [52]. SOLVED -$100
A set of integers $A$ is Ramsey $2$-complete if, whenever $A$ is $2$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of $A$.

Burr and Erdős [BuEr85] showed that there exists a constant $c>0$ such that it cannot be true that $\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2$ for all large $N$ and that there exists a Ramsey $2$-complete $A$ such that for all large $N$ $\lvert A\cap \{1,\ldots,N\}\rvert < (2\log_2N)^3.$ Improve either of these bounds.

The stated bounds are due to Burr and Erdős [BuEr85]. Resolved by Conlon, Fox, and Pham [CFP21], who constructed a Ramsey $2$-complete $A$ such that $\lvert A\cap \{1,\ldots,N\}\rvert \ll (\log N)^2$ for all large $N$.

SOLVED - $250 A set of integers$A$is Ramsey$r$-complete if, whenever$A$is$r$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of$A$. Prove any non-trivial bounds about the growth rate of such an$A$for$r>2$. A paper of Burr and Erdős [BuEr85] proves both upper and lower bounds for$r=2$, showing that there exists some$c>0$such that it cannot be true that $\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2$ for all large$N$, and also constructing a Ramsey$2$-complete$A$such that for all large$N$$\lvert A\cap \{1,\ldots,N\}\rvert \ll (\log N)^3.$ Burr has shown that the sequence of$k$th powers is Ramsey$r$-complete for every$r,k\geq 1$. Solved by Conlon, Fox, and Pham [CFP21], who constructed for every$r\geq 2$an$r$-Ramsey complete$A$such that for all large$N$$\lvert A\cap \{1,\ldots,N\}\rvert \ll r(\log N)^2,$ and showed that this is best possible, in that there exists some constant$c>0$such that if$A\subset \mathbb{N}$satisfies $\lvert A\cap \{1,\ldots,N\}\rvert \leq cr(\log N)^2$ for all large$N$then$A$cannot be$r$-Ramsey complete. See also [54]. Additional thanks to: Mehtaab Sawhney SOLVED Suppose$A\subseteq \{1,\ldots,N\}$is such that there are no$k+1$elements of$A$which are relatively prime. An example is the set of all multiples of the first$k$primes. Is this the largest such set? This was disproved for$k=212$by Ahlswede and Khachatrian [AhKh94], who suggest that their methods can disprove this for arbitrarily large$k$. Erdős later asked [Er95] if the conjecture remains true provided$N\geq (1+o(1))p_k^2$(or, in a weaker form, whether it is true for$N$sufficiently large depending on$k$). Additional thanks to: Zachary Chase SOLVED If$G$is a graph with infinite chromatic number and$a_1<a_2<\cdots $are lengths of the odd cycles of$G$then$\sum \frac{1}{a_i}=\infty$. Conjectured by Erdős and Hajnal, and solved by Liu and Montgomery [LiMo20]. In [Er81] Erdős asks whether the$a_i$must in fact have positive upper density. The lower density of the set can be$0$since there are graphs of arbitrarily large chromatic number and girth. See also [65]. SOLVED If$G$is a graph which contains odd cycles of$\leq k$different lengths then$\chi(G)\leq 2k+2$, with equality if and only if$G$contains$K_{2k+2}$. Conjectured by Bollobás and Erdős. Bollobás and Shelah have confirmed this for$k=1$. Proved by Gyárfás [Gy92], who proved the stronger result that, if$G$is 2-connected, then$G$is either$K_{2k+2}$or contains a vertex of degree at most$2k$. A stronger form was established by Gao, Huo, and Ma [GaHuMa21], who proved that if a graph$G$has chromatic number$\chi(G)\geq 2k+3$then$G$contains cycles of$k+1$consecutive odd lengths. Additional thanks to: David Penman SOLVED Is it true that the number of graphs on$n$vertices which do not contain$G$is $\leq 2^{(1+o(1))\mathrm{ex}(n;G)}?$ If$G$is not bipartite the answer is yes, proved by Erdős, Frankl, and Rödl [ErFrRo86]. The answer is no for$G=C_6$, the cycle on 6 vertices. Morris and Saxton [MoSa16] have proved there are at least $2^{(1+c)\mathrm{ex}(n;C_6)}$ such graphs for infinitely many$n$, for some constant$c>0$. It is still possible (and conjectured by Morris and Saxton) that the weaker bound of $2^{O(\mathrm{ex}(n;G))}$ holds for all$G$. Additional thanks to: Tuan Tran OPEN Does every graph on$n$vertices with$>\mathrm{ex}(n;C_4)$edges contain$\gg n^{1/2}$many copies of$C_4$? Conjectured by Erdős and Simonovits, who could not even prove that at least$2$copies of$C_4$are guaranteed. He, Ma, and Yang [HeMaYa21] have proved this conjecture when$n=q^2+q+1$for some even integer$q$. OPEN For any graph$H$is there some$c=c(H)>0$such that every graph$G$on$n$vertices that does not contain$H$as an induced subgraph contains either a complete graph or independent set on$\geq n^c$vertices? Conjectured by Erdős and Hajnal [ErHa89], who proved that a complete graph or independent set must exist on $\geq \exp(c_H\sqrt{\log n})$ many vertices, where$c_H>0$is some constant. This was improved by Bucić, Nguyen, Scott, and Seymour [BNSS23] to $\geq \exp(c_H\sqrt{\log n\log\log n}).$ OPEN If$G_1,G_2$are two graphs with chromatic number$\aleph_1$then must there exist a graph$G$whose chromatic number is$4$(or even$\aleph_0$) which is a subgraph of both$G_1$and$G_2$? Erdős, Hajnal, and Shelah have shown that$G_1$and$G_2$must both contain all sufficiently large cycles. OPEN Does every infinite graph with infinite chromatic number contain a cycle of length$2^n$for infinitely many$n$? Conjectured by Mihók and Erdős. It is likely that$2^n$can be replaced by any sufficiently quickly growing sequence (e.g. the squares). OPEN -$1000
Does every graph with minimum degree at least 3 contain a cycle of length $2^k$ for some $k\geq 2$?
Conjectured by Erdős and Gyárfás, who believed the answer must be negative, and in fact for every $r$ there must be a graph of minimum degree at least $r$ without a cycle of length $2^k$ for any $k\geq 2$.

This was solved in the affirmative if the minimum degree is larger than some absolute constant by Liu and Montgomery [LiMo20] (therefore disproving the above stronger conjecture of Erdős and Gyárfás). Liu and Montgomery prove a much stronger result: if the average degree of $G$ is sufficiently large then there is some large integer $\ell$ such that for every even integer $m\in [(\log \ell)^8,\ell]$, $G$ contains a cycle of length $m$.

OPEN
Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\cdots$ be the lengths of cycles in $G$. Is it true that $\sum\frac{1}{a_i}\gg \log k?$ Is the sum $\sum\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph?
A problem of Erdős and Hajnal. Gyárfás, Komlós, and Szemerédi [GyKoSz84] have proved that this sum is $\gg \log k$. Liu and Montgomery [LiMo20] have proved the asymptotically sharp lower bound of $\geq (\tfrac{1}{2}-o(1))\log k$.

OPEN - $500 Is there$A\subseteq \mathbb{N}$such that $\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}$ exists and is$\neq 0$? A suitably constructed random set has this property if we are allowed to ignore an exceptional set of density zero. The challenge is obtaining this with no exceptional set. Erdős believed the answer should be no. Erdős and Sárkzözy proved that $\frac{\lvert 1_A\ast 1_A(n)-\log n\rvert}{\sqrt{\log n}}\to 0$ is impossible. Erdős suggests it may even be true that the$\liminf$and$\limsup$of$1_A\ast 1_A(n)/\log n$are always separated by some absolute constant. SOLVED -$500
If $f:\mathbb{N}\to \{-1,+1\}$ then is it true that for every $C>0$ there exist $d,m\geq 1$ such that $\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > C?$
The Erdős discrepancy problem. This is true, and was proved by Tao [Ta16], who also proved the more general case when $f$ takes values on the unit sphere.

In [Er81] it is further conjectured that $\max_{md\leq x}\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert \gg \log x.$

OPEN
Is $\sum_{n\geq 2}\frac{1}{n!-1}$ irrational?
OPEN
Is $\sum_{n\geq 2}\frac{\omega(n)}{2^n}$ irrational? (Here $\omega(n)$ counts the number of distinct prime divisors of $n$.)
Erdős [Er48] proved that $\sum_n \frac{d(n)}{2^n}$ is irrational, where $d(n)$ is the divisor function.
OPEN
Let $\mathfrak{c}$ be the ordinal of the real numbers, $\beta$ be any countable ordinal, and $2\leq n<\omega$. Is it true that $\mathfrak{c}\to (\beta, n)_2^3$?
Erdős and Rado proved that $\mathfrak{c}\to (\omega+n,4)_2^3$ for any $2\leq n<\omega$.
SOLVED
Is it true that for every infinite arithmetic progression $P$ which contains even numbers there is some constant $c=c(P)$ such that every graph with average degree at least $c$ contains a cycle whose length is in $P$?
This has been proved by Bollobás [Bo77]. The best dependence of the constant $c(P)$ is unknown.

SOLVED - $100 Is there a set$A\subset \mathbb{N}$of density$0$and a constant$c>0$such that every graph on sufficiently many vertices with average degree$\geq c$contains a cycle whose length is in$A$? Bollobás [Bo77] proved that such a$c$does exist if$A$is an infinite arithmetic progression containing even numbers (see [71]). Erdős was 'almost certain' that if$A$is the set of powers of$2$then no such$c$exists (although he conjectured that$n$vertices and average degree$\gg (\log n)^{C}$suffices for some$C=O(1)$). If$A$is the set of squares (or the set of$p\pm 1$for$p$prime) then he had no guess. Solved by Verstraëte [Ve05], who gave a non-constructive proof that such a set$A$exists. Liu and Montgomery [LiMo20] proved that in fact this is true when$A$is the set of powers of$2$(more generally any set of even numbers which doesn't grow too quickly) - in particular this contradicts the previous belief of Erdős. Additional thanks to: Richard Montgomery SOLVED Let$k\geq 0$. Let$G$be a graph on$n$vertices such that every subgraph$H\subseteq G$contains an independent set of size$\geq \frac{1}{2}\lvert H\rvert-k$. Must$G$be the union of a bipartite graph and$O_k(1)$many vertices? Proved by Reed [Re99]. (Thanks also to Reed for pointing out that the case$k=0$is trivial, since if$G$is not bipartite then$G$contains an odd cycle.) OPEN -$500
Let $f(n)\to \infty$ (possibly very slowly). Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $f(n)$ edges?
Conjectured by Erdős, Hajnal, and Szemerédi [EHS82].

Rödl [Ro82] has proved this for hypergraphs, and also proved there is such a graph (with chromatic number $\aleph_0$) if $f(n)=\epsilon n$ for any fixed constant $\epsilon>0$.

It is open even for $f(n)=\sqrt{n}$. Erdős offered \$500 for a proof but only \$250 for a counterexample. This fails (even with $f(n)\gg n$) if the graph has chromatic number $\aleph_1$ (see [111]).

OPEN
Is there a graph of chromatic number $\aleph_1$ such that for all $\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices then $H$ contains an independent set of size $>n^{1-\epsilon}$?
Conjectured by Erdős, Hajnal, and Szemerédi [EHS82].

SOLVED
Is it true that in any $2$-colouring of the edges of $K_n$ there must exist at least $(1+o(1))\frac{n^2}{12}$ many edge-disjoint monochromatic triangles?
Conjectured by Erdős, Faudreee, and Ordman. This would be best possible, as witnessed by dividing the vertices of $K_n$ into two equal parts and colouring all edges between the parts red and all edges inside the parts blue.

The answer is yes, proved by Gruslys and Letzter [GrLe20].

In [Er97d] Erdős also asks for a lower bound for the count of edge-disjoint monochromatic triangles in single colour (the colour chosen to maximise this quantity), and speculates that the answer is $\geq cn^2$ for some constant $c?1/24$.

OPEN - $250 Find the value of$\lim_{k\to \infty}R(k)^{1/k}$. Erdős offered \$100 for just a proof of the existence of this constant, without determining its value. He also offered \$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. Erdős proved $\sqrt{2}\leq \liminf_{k\to \infty}R(k)^{1/k}\leq \limsup_{k\to \infty}R(k)^{1/k}\leq 4.$ The upper bound has been improved to$4-\tfrac{1}{128}$by Campos, Griffiths, Morris, and Sahasrabudhe [CGMS23]. This problem is #3 in Ramsey Theory in the graphs problem collection. OPEN -$100
Give a constructive proof that $R(k)>C^k$ for some constant $C>1$.
Erdős gave a simple probabilistic proof that $R(k) \gg k2^{k/2}$. Equivalently, this question asks for an explicit construction of a graph on $n$ vertices which does not contain any clique or independent set of size $\geq c\log n$ for some constant $c>0$. Cohen [Co15] (see the introduction for further history) constructed a graph on $n$ vertices which does not contain any clique or independent set of size $\geq 2^{(\log\log n)^{C}}$ for some constant $C>0$. Li [Li23b] has recently improved this to $\geq (\log n)^{C}$ for some constant $C>0$.

This problem is #4 in Ramsey Theory in the graphs problem collection.

Additional thanks to: Jesse Goodman, Mehtaab Sawhney
OPEN
We say $G$ is Ramsey size linear if $R(G,H)\ll m$ for all graphs $H$ with $m$ edges and no isolated vertices.

Are there infinitely many graphs $G$ which are not Ramsey size linear but such that all of its subgraphs are?

Asked by Erdős, Faudree, Rousseau, and Schelp [EFRS93]. $K_4$ is the only known example of such a graph.
OPEN
Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is contained in at least one triangle, must contain an edge in at least $m$ different triangles. Estimate $f_c(n)$. In particular, is it true that $f_c(n)>n^{\epsilon}$ for some $\epsilon>0$? Or even $f_c(n)\gg \log n$?
A problem of Erdős and Rothschild. Alon and Trotter showed that $f_c(n)\ll_c n^{1/2}$. Szemerédi observed that his regularity lemma implies that $f_c(n)\to \infty$.

OPEN
Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $n^2/6+O(n)$ many cliques?
Asked by Erdős, Ordman, and Zalcstein [EOZ93], who proved an upper bound of $(1/4-\epsilon)n^2$ many cliques (for some very small $\epsilon>0$). The example of all edges between a complete graph on $n/3$ vertices and an empty graph on $2n/3$ vertices show that $n^2/6+O(n)$ is sometimes necessary.

A split graph is one where the vertices can be split into a clique and an independent set. Every split graph is chordal. Chen, Erdős, and Ordman [CEO94] have shown that any split graph can be partitioned into $\frac{3}{16}n^2+O(n)$ many cliques.

OPEN
Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $F(n)/\log n\to \infty$.
Conjectured by Erdős, Fajtlowicz, and Stanton. It is known that $F(5)=3$ and $F(7)=4$. Ramsey's theorem implies that $F(n)\gg \log n$. Bollobás observed that $F(n)\ll n^{1/2+o(1)}$. Alon, Krivelevich, and Sudakov [AKS07] have improved this to $n^{1/2}(\log n)^{O(1)}$.
SOLVED - $500 Suppose that we have a family$\mathcal{F}$of subsets of$[4n]$such that$\lvert A\rvert=2n$for all$A\in\mathcal{F}$and for every$A,B\in \mathcal{F}$we have$\lvert A\cap B\rvert \geq 2$. Then $\lvert \mathcal{F}\rvert \leq \frac{1}{2}\left(\binom{4n}{2n}-\binom{2n}{n}^2\right).$ Conjectured by Erdős, Ko, and Rado [ErKoRa61]. This inequality would be best possible, as shown by taking$\mathcal{F}$to be the collection of all subsets of$[4n]$of size$2n$containing at least$n+1$elements from$[2n]$. Proved by Ahlswede and Khachatrian [AhKh97], who more generally showed the following. Let$2\leq t\leq k\leq m$and let$r\geq 0$be such that $\frac{1}{r+1}\leq \frac{m-2k+2t-2}{(t-1)(k-t+1)}< \frac{1}{r}.$ The largest possible family of subsets of$[m]$of size$k$, such that the pairwise intersections have size at least$t$, is the family of all subsets of$[m]$of size$k$which contain at least$t+r$elements from$\{1,\ldots,t+2r\}$. Additional thanks to: Tuan Tran OPEN The cycle set of a graph$G$on$n$vertices is a set$A\subseteq \{3,\ldots,n\}$such that there is a cycle in$G$of length$\ell$if and only if$\ell \in A$. Let$f(n)$count the number of possible such$A$. Prove that$f(n)=o(2^n)$. Prove that$f(n)/2^{n/2}\to \infty$. Conjectured by Erdős and Faudree, who showed that$2^{n/2}<f(n) \leq 2^{n-2}$. The first problem was solved by Verstraëte [Ve04], who proved $f(n)\ll 2^{n-n^c}$ for some constant$c>0$. One can also ask about the existence and value of$\lim f(n)^{1/n}$. Additional thanks to: Tuan Tran OPEN Let$f(n)$be such that every graph on$n$vertices with minimal degree$\geq f(n)$contains a$C_4$. Is it true that$f(n+1)\geq f(n)$? A weaker version of the conjecture asks for some constant$c$such that$f(m)>f(n)-c$for all$m>n$. This question can be asked for other graphs than$C_4$. OPEN -$100
Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that every subgraph of $Q_n$ with $\geq \left(\frac{1}{2}+o(1)\right)n2^{n-1}$ many edges contains a $C_4$?
The best known result is due to Balogh, Hu, Lidicky, and Liu [BHLL14], who proved that $0.6068 n2^{n-1}$ edges suffice.

A similar question can be asked for other even cycles.

OPEN
Let $\epsilon >0$. Is it true that, if $k$ is sufficiently large, then $R(G)>(1-\epsilon)^kR(k)$ for every graph $G$ with chromatic number $\chi(G)=k$?

Even stronger, is there some $c>0$ such that, for all large $k$, $R(G)>cR(k)$ for every graph $G$ with chromatic number $\chi(G)=k$?

Erdős originally conjectured that $R(G)\geq R(k)$, which is trivial for $k=3$, but fails already for $k=4$, as Faudree and McKay [FaMc93] showed that $R(W)=17$ for the pentagonal wheel $W$.

Since $R(k)\leq 4^k$ this is trivial for $\epsilon\geq 3/4$. Yuval Wigderson points out that $R(G)\gg 2^{k/2}$ for any $G$ with chromatic number $k$ (via a random colouring), which asymptotically matches the best-known lower bounds for $R(k)$.

This problem is #12 and #13 in Ramsey Theory in the graphs problem collection.

SOLVED - $100 For any$\epsilon>0$there exists$\delta=\delta(\epsilon)>0$such that if$G$is a graph on$n$vertices with no independent set or clique of size$\geq \epsilon\log n$then$G$contains an induced subgraph with$m$edges for all$m\leq \delta n^2$. Conjectured by Erdős and McKay, who proved it with$\delta n^2$replaced by$\delta (\log n)^2$. Solved by Kwan, Sah, Sauermann, and Sawhney [KSSS22]. Erdős' original formulation also had the condition that$G$has$\gg n^2$edges, but an old result of Erdős and Szemerédi says that this follows from the other condition anyway. Additional thanks to: Zachary Hunter and Mehtaab Sawhney OPEN -$500
Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances?
A $\sqrt{n}\times\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz [GuKa15] who proved that there are always $\gg n/\log n$ many distinct distances.

A stronger form (see [604]) may be true: is there a single point which determines $\gg n/\sqrt{\log n}$ distinct distances, or even $\gg n$ many such points, or even that this is true averaged over all points.

OPEN - $500 Does every set of$n$distinct points in$\mathbb{R}^2$contain at most$n^{1+O(1/\log\log n)}$many pairs which are distance 1 apart? The unit distance problem. In [Er94b] Erdős dates this conjecture to 1946. This would be the best possible, as is shown by a set of lattice points. It is easy to show that there are$O(n^{3/2})$many such pairs. The best known upper bound is$O(n^{4/3})$, due to Spencer, Szemerédi, and Trotter [SST84]. In [Er83c] and [Er85] Erdős offers \$250 for an upper bound of the form $n^{1+o(1)}$.

Part of the difficulty of this problem is explained by a result of Valtr (see [Sz16]), who constructed a metric on $\mathbb{R}^2$ and a set of $n$ points with $\gg n^{4/3}$ unit distance pairs (with respect to this metric). The methods of the upper bound proof of Spencer, Szemerédi, and Trotter [SST84] generalise to include this metric. Therefore to prove an upper bound better than $n^{4/3}$ some special feature of the Euclidean metric must be exploited.

See a survey by Szemerédi [Sz16] for further background and related results.

OPEN
Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that for large $n$ there are at least two (and probably many) such $A$ which are non-similar.
For $n=5$ the regular pentagon is the unique such set (Erdős mysteriously remarks this was proved by 'a colleague').
OPEN - $500 Let$f(n)$be maximal such that there exists a set$A$of$n$points in$\mathbb{R}^2$in which every$x\in A$has at least$f(n)$points in$A$equidistant from$x$. Is it true that$f(n)\leq n^{o(1)}$? Or even$f(n) < n^{c/\log\log n}$for some constant$c>0$? This is a stronger form of the unit distance conjecture (see [90]). The set of lattice points imply$f(n) > n^{c/\log\log n}$for some constant$c>0$. Erdős offered \$500 for a proof that $f(n) \leq n^{o(1)}$ but only \$100 for a counterexample. It is trivial that$f(n) \ll n^{1/2}$. A result of Pach and Sharir implies$f(n) \ll n^{2/5}$. Fishburn (personal communication to Erdős) proved that$6$is the smallest$n$such that$f(n)=3$and$8$is the smallest$n$such that$f(n)=4$, and suggested that the lattice points may not be best example. See also [754]. SOLVED If$n$distinct points in$\mathbb{R}^2$form a convex polygon then they determine at least$\lfloor \frac{n+1}{2}\rfloor$distinct distances. Solved by Altman [Al63]. The stronger variant that says there is one point which determines at least$\lfloor \frac{n+1}{2}\rfloor$distinct distances is still open. Fishburn in fact conjectures that if$R(x)$counts the number of distinct distances from$x$then $\sum_{x\in A}R(x) \geq \binom{n}{2}.$ Szemerédi conjectured (see [Er97e]) that this stronger variant remains true if we only assume that no three points are on a line, and proved this with the weaker bound of$n/3$. See also [660]. SOLVED Suppose$n$points in$\mathbb{R}^2$determine a convex polygon and the set of distances between them is$\{u_1,\ldots,u_t\}$. Suppose$u_i$appears as the distance between$f(u_i)$many pairs of points. Then $\sum_i f(u_i)^2 \ll n^3.$ Solved by Fishburn [Al63]. Note it is trivial that$\sum f(u_i)=\binom{n}{2}$. The stronger conjecture that$\sum f(u_i)^2$is maximal for the regular$n$-gon (for large enough$n$) is still open. See also [95]. SOLVED -$500
Let $x_1,\ldots,x_n\in\mathbb{R}^2$ determine the set of distances $\{u_1,\ldots,u_t\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then for all $\epsilon>0$ $\sum_i f(u_i)^2 \ll_\epsilon n^{3+\epsilon}.$
The case when the points determine a convex polygon was been solved by Fishburn [Al63]. Note it is trivial that $\sum f(u_i)=\binom{n}{2}$. Solved by Guth and Katz [GuKa15] who proved the upper bound $\sum_i f(u_i)^2 \ll n^3\log n.$

OPEN
If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.
Conjectured by Erdős and Moser. Füredi [Fu90] proved an upper bound of $O(n\log n)$. A short proof of this bound was given by Brass and Pach [BrPa01]. The best known upper bound is $\leq n\log_2n+4n,$ due to Aggarwal [Ag15].

Edelsbrunner and Hajnal [EdHa91] have constructed $n$ such points with $2n-7$ pairs distance $1$ apart. (This disproved an early stronger conjecture of Erdős and Moser, that the true answer was $\frac{5}{3}n+O(1)$.)

OPEN - $100 Does every convex polygon have a vertex with no other$4$vertices equidistant from it? Erdős originally conjectured this with no$3$vertices equidistant, but Danzer found a convex polygon on 9 points such that every vertex has three vertices equidistant from it (but this distance depends on the vertex), and Fishburn and Reeds [FiRe92] have found a convex polygon on 20 points such that every vertex has three vertices equidistant from it (and this distance is the same for all vertices). If this fails for$4$, perhaps there is some constant for which it holds? Erdős suggested this as an approach to solve [96]. Indeed, if this problem holds for$k+1$vertices then, by induction, this implies an upper bound of$kn$for [96]. The answer is no if we omit the requirement that the polygon is convex (I thank Boris Alexeev and Dustin Mixon for pointing this out), since for any$d$there are graphs with minimum degree$d$which can be embedded in the plane such that each edge has length one (for example one can take the$d$-dimensional hypercube graph on$2^d$vertices). One can then connect the vertices in a cyclic order so that there are no self-intersections and no three consecutive vertices on a line, thus forming a (non-convex) polygon. Additional thanks to: Boris Alexeev and Dustin Mixon OPEN Let$h(n)$be such that any$n$points in$\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least$h(n)$distinct distances. Does$h(n)/n\to \infty$? Erdős could not even prove$h(n)\geq n$. Pach has shown$h(n)<n^{\log_23}$. Erdős, Füredi, and Pach have improved this to $h(n) < n\exp(c\sqrt{\log n})$ for some constant$c>0$. OPEN -$100
Let $A\subseteq\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently large then must there be three points in $A$ which form an equilateral triangle of size 1?
Thue proved that the minimal such diameter is achieved (asymptotically) by the points in a triangular lattice intersected with a circle. In general Erdős believed such a set must have very large intersection with the triangular lattice (perhaps as many as $(1-o(1))n$).

Erdős [Er94b] wrote 'I could not prove it but felt that it should not be hard. To my great surprise both B. H. Sendov and M. Simonovits doubted the truth of this conjecture.' In [Er94b] he offers \$100 for a counterexample but only \$50 for a proof.

The stated problem is false for $n=4$, for example taking the points to be vertices of a square. The behaviour of such sets for small $n$ is explored by Bezdek and Fodor [BeFo99].

Additional thanks to: Boris Alexeev and Dustin Mixon
OPEN
Let $A$ be a set of $n$ points in $\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\gg n$?
Perhaps the diameter is even $\geq n-1$ for sufficiently large $n$. Piepmeyer has an example of $9$ such points with diameter $<5$. Kanold proved the diameter is $\geq n^{3/4}$. The bounds on the distinct distance problem [89] proved by Guth and Katz [GuKa15] imply a lower bound of $\gg n/\log n$.
Additional thanks to: Shengtong Zhang, Boris Alexeev and Dustin Mixon
OPEN - $100 Given$n$points in$\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is$o(n^2)$. There are examples of sets of$n$points with$\sim n^2/6$many collinear triples and no four points on a line. Such constructions are given by Burr, Grünbaum, and Sloane [BGS74] and Füredi and Palásti [FuPa84]. Grünbaum [Gr76] constructed an example with$\gg n^{3/2}$such lines. Erdős speculated this may be the correct order of magnitude. This is false: Solymosi and Stojaković [SoSt13] have constructed a set with no five on a line and at least $n^{2-O(1/\sqrt{\log n})}$ many lines containing exactly four points. See also [102]. A generalisation of this problem is asked in [588]. Additional thanks to: Zach Hunter OPEN Let$c>0$and$h_c(n)$be such that for any$n$points in$\mathbb{R}^2$such that there are$\geq cn^2$lines each containing more than three points, there must be some line containing$h_c(n)$many points. Estimate$h_c(n)$. Is it true that, for fixed$c>0$, we have$h_c(n)\to \infty$? A problem of Erdős and Purdy. It is not even known if$h_c(n)\geq 5$(see [101]). It is easy to see that$h_c(n) \ll_c n^{1/2}$, and Erdős originally suggested that perhaps a similar lower bound$h_c(n)\gg_c n^{1/2}$holds. Zach Hunter has pointed out that this is false, even replacing$>3$points on each line with$>k$points: consider the set of points in$\{1,\ldots,m\}^d$where$n\approx m^d$. These intersect any line in$\ll_d n^{1/d}$points, and have$\gg_d n^2$many pairs of points each of which determine a line with at least$k$points. This is a construction in$\mathbb{R}^d$, but a random projection into$\mathbb{R}^2$preserves the relevant properties. This construction shows that$h_c(n) \ll n^{1/\log(1/c)}$. Additional thanks to: Zach Hunter OPEN Let$h(n)$count the number of incongruent sets of$n$points in$\mathbb{R}^2$which minimise the diameter subject to the constraint that$d(x,y)\geq 1$for all points$x\neq y$. Is it true that$h(n)\to \infty$? It is not even known whether$h(n)\geq 2$for all large$n$. See also [99]. OPEN Given$n$points in$\mathbb{R}^2$the number of distinct unit circles containing at least three points is$o(n^2)$. In [Er81d] Erdős proved that$\gg n$many circles is possible, and that there cannot be more than$n(n-1)$many circles. Elekes [El84] has a simple construction of a set with$\gg n^{3/2}$such circles. This may be the correct order of magnitude. OPEN -$50
Let $A,B\subset \mathbb{R}^2$ be disjoint sets of size $n$ and $n-3$ respectively, with not all of $A$ contained on a single line. Is there a line which contains at least two points from $A$ and no points from $B$?
Conjectured by Erdős and Purdy [ErPu95] (the prize is for a proof or disproof). A construction of Hickerson shows that this fails with $n-2$. A result independently proved by Beck [Be83] and Szemerédi and Trotter [SzTr83] (see [211]) implies it is true with $n-3$ replaced by $cn$ for some constant $c>0$.
OPEN
Draw $n$ squares inside the unit square with no common interior point. Let $f(n)$ be the maximum possible total perimeter of the squares. Is $f(k^2+1)=4k$?
In [Er94b] Erdős dates this conjecture to 'more than 60 years ago'.

It is trivial from the Cauchy-Schwarz inequality that $f(k^2)=4k$. Erdős also asks for which $n$ is it true that $f(n+1)=f(n)$.

OPEN - $500 Let$f(n)$be minimal such that any$f(n)$points in$\mathbb{R}^2$, no three on a line, contain$n$points which form the vertices of a convex$n$-gon. Prove that$f(n)=2^{n-2}+1$. The Erdős-Klein-Szekeres 'Happy Ending' problem. The problem originated in 1931 when Klein observed that$f(4)=5$. Turán and Makai showed$f(5)=9$. Erdős and Szekeres proved the bounds $2^{n-2}+1\leq f(n)\leq \binom{2n-4}{n-2}+1.$ ([ErSz60] and [ErSz35] respectively). There were several improvements of the upper bound, but all of the form$4^{(1+o(1))n}$, until Suk [Su17] proved $f(n) \leq 2^{(1+o(1))n}.$ The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20], who prove $f(n) \leq 2^{n+O(\sqrt{n\log n})}.$ In [Er97e] Erdős clarifies that the \$500 is for a proof, and only offers \$100 for a disproof. This problem is #1 in Ramsey Theory in the graphs problem collection. See also [216] and [651]. Additional thanks to: Casey Tompkins OPEN For every$r\geq 4$and$k\geq 2$is there some finite$f(k,r)$such that every graph of chromatic number$\geq f(k,r)$contains a subgraph of girth$\geq r$and chromatic number$\geq k$? Conjectured by Erdős and Hajnal. Rödl [Ro77] has proved the$r=4$case. The infinite version (whether every graph of infinite chromatic number contains a subgraph of infinite chromatic number whose girth is$>k$) is also open. In [Er79b] Erdős also asks whether $\lim_{k\to \infty}\frac{f(k,r+1)}{f(k,r)}=\infty.$ SOLVED Any$A\subseteq \mathbb{N}$of positive upper density contains a sumset$B+C$where both$B$and$C$are infinite. The Erdős sumset conjecture. Proved by Moreira, Richter, and Robertson [MRR19]. Additional thanks to: Antonio Girao OPEN Is there some$F(n)$such that every graph with chromatic number$\aleph_1$has, for all large$n$, a subgraph with chromatic number$n$on at most$F(n)$vertices? Conjectured by Erdős, Hajnal, and Szemerédi [EHS82]. This fails if the graph has chromatic number$\aleph_0$. OPEN If$G$is a graph let$h_G(n)$be defined such that any subgraph of$G$on$n$vertices can be made bipartite after deleting at most$h_G(n)$edges. What is the behaviour of$h_G(n)$? Is it true that$h_G(n)/n\to \infty$for every graph$G$with chromatic number$\aleph_1$? A problem of Erdős, Hajnal, and Szemerédi [EHS82]. Every$G$with chromatic number$\aleph_1$must have$h_G(n)\gg n$since$G$must contain, for some$r$,$\aleph_1$many vertex disjoint odd cycles of length$2r+1$. On the other hand, Erdős, Hajnal, and Szemerédi proved that there is a$G$with chromatic number$\aleph_1$such that$h_G(n)\ll n^{3/2}$. In [Er81] Erdős conjectures that this can be improved to$\ll n^{1+\epsilon}$for every$\epsilon>0$. See also [74]. OPEN Let$k=k(n,m)$be minimal such that any directed graph on$k$vertices must contain either an independent set of size$n$or a directed path of size$m$. Determine$k(n,m)$. A problem of Erdős and Rado [ErRa67], who showed $k(n,m) \leq \frac{2^{m-1}(n-1)^m+n-2}{2n-3}.$ Larson and Mitchell [LaMi97] prove that$k(n,m)\leq n^{m-1}$for$m\geq 3$. SOLVED -$250
If $G$ is bipartite then $\mathrm{ex}(n;G)\ll n^{3/2}$ if and only $G$ is $2$-degenerate, that is, $G$ contains no induced subgraph with minimal degree at least 3.
Conjectured by Erdős and Simonovits. Erdős offered \$250 for a proof and \$100 for a counterexample. Disproved by Janzer [Ja21], who constructed, for any $\epsilon>0$, a $3$-regular bipartite graph $H$ such that $\mathrm{ex}(n;H)\ll n^{\frac{4}{3}+\epsilon}.$

See also [146] and [147] and the entry in the graphs problem collection.

OPEN
If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\{ z\in \mathbb{C} : \lvert p(z)\rvert=1\}$ maximised when $p(z)=z^n-1$?
A problem of Erdős, Herzog, and Piranian [EHP58].
SOLVED
If $p(z)$ is a polynomial of degree $n$ such that $\{z : \lvert p(z)\rvert\leq 1\}$ is connected then is it true that $\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?$
The lower bound is easy: this is $\geq n$ and equality holds if and only if $p(z)=z^n$. The assumption that the set is connected is necessary, as witnessed for example by $p(z)=z^2+10z+1$.

The Chebyshev polynomials show that $n^2/2$ is best possible here. Erdős originally conjectured this without the $o(1)$ term but Szabados observed that was too strong. Pommerenke [Po59a] proved an upper bound of $\frac{e}{2}n^2$.

Eremenko and Lempert [ErLe94] have shown this is true, and in fact Chebyshev polynomials are the extreme examples.

SOLVED
Let $p(z)=\prod_{i=1}^n (z-z_i)$ for $\lvert z_i\rvert \leq 1$. Then the area of the set where $A=\{ z: \lvert p(z)\rvert <1\}$ is $>n^{-O(1)}$ (or perhaps even $>(\log n)^{-O(1)}$).
Conjectured by Erdős, Herzog, and Piranian [ErHePi58]. The lower bound $\mu(A) \gg n^{-4}$ follows from a result of Pommerenke [Po61]. The stronger lower bound $\gg (\log n)^{-O(1)}$ is still open.

Wagner [Wa88] proves, for $n\geq 3$, the existence of such polynomials with $\mu(A) \ll_\epsilon (\log\log n)^{-1/2+\epsilon}$ for all $\epsilon>0$.

Additional thanks to: Boris Alexeev and Dustin Mixon
OPEN
Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.

Estimate $h(n)$ as well as possible.

Pyber [Py87] has proved there exist constants $c_2>c_1>1$ such that $c_1^n<h(n)<c_2^n$. Erdős [Er97f] writes that the lower bound was already known to Isaacs.
SOLVED
Let $\alpha$ be a cardinal or ordinal number or an order type such that every two-colouring of $K_\alpha$ contains either a red $K_\alpha$ or a blue $K_3$. For every $n\geq 3$ must every two-colouring of $K_\alpha$ contain either a red $K_\alpha$ or a blue $K_n$?
Conjectured by Erdős and Hajnal. In arrow notation, this is asking where $\alpha \to (\alpha,3)^2$ implies $\alpha \to (\alpha, n)^2$ for every finite $n$.

The answer is no, as independently shown by Schipperus [Sc99] (published in [Sc10]) and Darby [Da99].

For example, Larson [La00] has shown that this is false when $\alpha=\omega^{\omega^2}$ and $n=5$. There is more background and proof sketches in Chapter 2.9 of [HST10], by Hajnal and Larson.

Additional thanks to: Zachary Chase, Andrés Caicedo
SOLVED - $100 Let$z_i$be an infinite sequence of complex numbers such that$\lvert z_i\rvert=1$for all$i\geq 1$, and for$n\geq 1$let $p_n(z)=\prod_{i\leq n} (z-z_i).$ Let$M_n=\max_{\lvert z\rvert=1}\lvert p_n(z)\rvert$. Is it true that$\limsup M_n=\infty$? Is it true that there exists$c>0$such that for infinitely many$n$we have$M_n > n^c$, or even that for all$n$$\sum_{k\leq n}M_k > n^{1+c}?$ The weaker conjecture that$\limsup M_n=\infty$was proved by Wagner, who show that there is some$c>0$with$M_n>(\log n)^c$infinitely often. This was solved by Beck [Be91], who proved that there exists some$c>0$such that $\max_{n\leq N} M_n > N^c.$ Additional thanks to: Winston Heap OPEN -$100
Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\in\mathbb{R}$ and $a\neq 0$?
The Erdős similarity problem.

This is true if $A$ is unbounded or dense in some interval. It therefore suffices to prove this when $A=\{a_1>a_2>\cdots\}$ is a countable strictly monotone sequence which converges to $0$.

Steinhaus [St20] has proved this is false whenever $A$ is a finite set.

This conjecture is known in many special cases (but, for example, it is is open when $A=\{1,1/2,1/4,\ldots\}$. For an overview of progress we recommend a nice survey by Svetic [Sv00] on this problem.

SOLVED
Let $F_{k}(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that the product of no $k$ many distinct elements of $A$ is a square. Is $F_5(N)=(1-o(1))N$? More generally, is $F_{2k+1}(N)=(1-o(1))N$?
Conjectured by Erdős, Sós, and Sárkzözy [ESS95], who proved $F_2(N)=\left(\frac{6}{\pi^2}+o(1)\right)N,$ $F_3(N) = (1-o(1))N,$ and also established asymptotics for $F_k(N)$ for all even $k\geq 4$ (for which $F_k(N)\asymp N/\log N$ for all even $k\geq 4$. Erdős [Er38] earlier proved that $F_4(N)=o(N)$ (indeed, if $\lvert A\rvert \gg N$ and $A\subseteq \{1,\ldots,N\}$ then there is a non-trivial solution to $ab=cd$ with $a,b,c,d\in A$.)

Erdős (and independently Hall [Ha96] and Montgomery) also asked about $F(N)$, the size of the largest $A\subseteq\{1,\ldots,N\}$ such that the product of no odd number of $a\in A$ is a square. Ruzsa [Ru77] observed that $1/2<\lim F(N)/N <1$. Granville and Soundararajan [GrSo01] proved an asymptotic $F(N)=(1-c+o(1))N$ where $c=0.1715\ldots$ is an explicit constant.

This problem was answered in the negative by Tao [Ta24], who proved that for any $k\geq 4$ there is some constant $c_k>0$ such that $F_k(N) \leq (1-c_k+o(1))N$.

OPEN
Let $f(n)$ be a number theoretic function which grows slowly (e.g. slower than $(\log n)^{1-c}$) and $F(n)$ be such that for almost all $n$ we have $f(n)/F(n)\to 0$. When are there infinitely many $x$ such that $\frac{\#\{ n\in \mathbb{N} : n+f(n)\in (x,x+F(x))\}}{F(x)}\to \infty?$
Conjectured by Erdős, Pomerance, and Sárközy [ErPoSa97] who prove this when $f$ is the divisor function or the number of distinct prime divisors of $n$, but Erdős believed it is false when $f(n)=\phi(n)$ or $\sigma(n)$.
OPEN - $250 Let$a,b,c$be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form$a^kb^lc^m$($k,l,m\geq 0$), none of which divide any other? Conjectured by Erdős and Lewin [ErLe96], who (among other related results) prove this when$a=3$,$b=5$, and$c=7$. OPEN Let$3\leq d_1<d_2<\cdots <d_k$be integers such that $\sum_{1\leq i\leq k}\frac{1}{d_i-1}\geq 1.$ Can all sufficiently large integers be written as a sum of the shape$\sum_i c_ia_i$where$c_i\in \{0,1\}$and$a_i$has only the digits$0,1$when written in base$d_i$? Conjectured by Burr, Erdős, Graham, and Li [BEGL96]. Pomerance observed that the condition$\sum 1/(d_i-1)\geq 1$is necessary. In [BEGL96] they prove the property holds for$\{3,4,7\}$. See also [125]. Additional thanks to: Boris Alexeev and Dustin Mixon OPEN Let$A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$be the set of integers which have only the digits$0,1$when written base$3$, and$B=\{ \sum\epsilon_k4^k : \epsilon_k\in \{0,1\}\}$be the set of integers which have only the digits$0,1$when written base$4$. Does$A+B$have positive density? A problem of Burr, Erdős, Graham, and Li [BEGL96]. More generally, if$n_1<\cdots<n_k$have $\sum_{i=1}^k\log_{n_k}(2)>1$ and$A_i$is the set of integers with only the digits$0,1$in base$n_i$then does$A_1+\cdots+A_k$have positive density? See also [124]. OPEN -$250
Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a\neq b\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\log n\to\infty$?
Investigated by Erdős and Turán [ErTu34] (prompted by a question of Lázár and Grünwald) in their first joint paper, where they proved that $\log n \ll f(n) \ll n/\log n$ (the upper bound is trivial, taking $A=\{1,\ldots,n\}$). Erdős says that $f(n)=o(n/\log n)$ has never been proved, but perhaps never seriously attacked.
SOLVED
Let $f(m)$ be maximal such that every graph with $m$ edges must contain a bipartite graph with $\geq \frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}+f(m)$ edges. Is there an infinite sequence of $m_i$ such that $f(m_i)\to \infty$?
Conjectured by Erdős, Kohayakava, and Gyárfás. Edwards [Ed73] proved that $f(m)\geq 0$ always. Note that $f(\binom{n}{2})= 0$, taking $K_n$. Solved by Alon [Al96], who showed $f(n^2/2)\gg n^{1/2}$, and also showed that $f(m)\ll m^{1/4}$ for all $m$. The best possible constant in $f(m)\leq Cm^{1/4}$ is unknown.
OPEN - $250 Let$G$be a graph with$n$vertices such that every subgraph on$\geq n/2$vertices has more than$n^2/50$edges. Must$G$contain a triangle? A problem of Erdős and Rousseau. The constant$50$would be best possible as witnessed by a blow-up of$C_5$or the Petersen graph. Krivelevich [Kr95] has proved this with$n/2$replaced by$3n/5$(and$50$replaced by$25$). Keevash and Sudakov [KeSu06] have proved this under the additional assumption that$G$has at most$n^2/12$edges. Additional thanks to: Boris Alexeev OPEN Let$R(n;k,r)$be the smallest$N$such that if the edges of$K_N$are$r$-coloured then there is a set of$n$vertices which does not contain a copy of$K_k$in at least one of the$r$colours. Prove that there is a constant$C=C(r)>1$such that $R(n;3,r) < C^{\sqrt{n}}.$ Conjectured by Erdős and Gyárfás, who proved the existence of some$C>1$such that$R(n;3,r)>C^{\sqrt{n}}$. Note that when$r=k=2$we recover the classic Ramsey numbers. Erdős thought it likely that for all$r,k\geq 2$there exists some$C_1,C_2>1$(depending only on$r$) such that $C_1^{n^{1/k-1}}< R(n;k,r) < C_2^{n^{1/k-1}}.$ Antonio Girao has pointed out that this problem as written is easily disproved, and indeed$R(n;3,2) \geq C^{n}$: The obvious probabilistic construction (randomly colour the edges red/blue independently uniformly at random) yields a 2-colouring of the edges of$K_N$such every set on$n$vertices contains a red triangle and a blue triangle (using that every set of$n$vertices contains$\gg n^2$edge-disjoint triangles), provided$N \leq C^n$for some absolute constant$C>1$. This implies$R(n;3,2) \geq C^{n}$, contradicting the conjecture. Perhaps Erdős had a different problem in mind, but it is not clear what that might be. It would presumably be one where the natural probabilistic argument would deliver a bound like$C^{\sqrt{n}}$as Erdős and Gyárfás claim to have achieved via the probabilistic method. Additional thanks to: Antonio Girao OPEN Let$A\subset\mathbb{R}^2$be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in$A$, where two vertices are joined by an edge if and only if they are an integer distance apart. How large can the chromatic number and clique number of this graph be? In particular, can the chromatic number be infinite? Asked by Andrásfai and Erdős. Erdős [Er97b] also asked where such a graph could contain an infinite complete graph, but this is impossible by an earlier result of Anning and Erdős [AnEr45]. See also [213]. Additional thanks to: Noga Alon OPEN Let$\epsilon>0$and$N$be sufficiently large depending on$\epsilon$. Is there$A\subseteq\{1,\ldots,N\}$such that no$a\in A$divides the sum of any distinct elements of$A\backslash\{a\}$and$\lvert A\rvert\gg N^{1/2-\epsilon}$? It is easy to see that we must have$\lvert A\rvert \ll N^{1/2}$. Csaba has constructed such an$A$with$\lvert A\rvert \gg N^{1/5}$. OPEN -$100
Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points? Must the number of such distances $\to \infty$ as $n\to \infty$?
Asked by Erdős and Pach. Hopf and Pannowitz [HoPa34] proved that the largest distance between points of $A$ can occur at most $n$ times, but it is unknown whether a second such distance must occur.

It may be true that there are at least $n^{1-o(1)}$ many such distances. In [Er97e] Erdős offers \$100 for 'any nontrivial result'. See also [223] and [756]. SOLVED Let$f(n)$be minimal such that every triangle-free graph$G$with$n$vertices and diameter$2$contains a vertex with degree$\geq f(n)$. What is the order of growth of$f(n)$? Does$f(n)/\sqrt{n}\to \infty$? Asked by Erdős and Pach. The lower bound$f(n)\geq (1-o(1))\sqrt{n}$follows from the fact that a graph with maximum degree$d$and diameter$2$has at most$1+d+d(d-1)=d^2+1$many vertices. Simonovits observed that the subsets of$[3m-1]$of size$m$, two sets joined by edge if and only if they are disjoint, forms a triangle-free graph of diameter$2$which is regular of degree$\binom{2m-1}{m}$. This construction proves that $f(n) \leq n^{(1+o(1))\frac{2}{3H(1/3)}}=n^{0.7182\cdots},$ where$H(x)$is the binary entropy function. In [Er97b] Erdős encouraged the reader to try and find a better construction. In this note Alon provides a simple construction that proves$f(n) \ll \sqrt{n\log n}$: take a triangle-free graph with independence number$\ll \sqrt{n\log n}$(the existence of which is the lower bound in [165]) and add edges until it has diameter$2$; the neighbourhood of any set is an independent set and hence the maximum degree is still$\ll \sqrt{n\log n}$. Hanson and Seyffarth [HaSe84] proved that$f(n)\leq (\sqrt{2}+o(1))\sqrt{n}$using a Cayley graph on$\mathbb{Z}/n\mathbb{Z}$, with the generating set given by some symmetric complete sum-free set of size$\sim \sqrt{n}$. An alternative construction of such a complete sum-free set was given by Haviv and Levy [HaLe18]. Füredi and Seress [FuSe94] proved that$f(n)\leq (\frac{2}{\sqrt{3}}+o(1))\sqrt{n}$. The precise asymptotics of$f(n)$are unknown; Alon believes that the truth is$f(n)\sim \sqrt{n}$. Additional thanks to: Noga Alon, Ishay Haviv SOLVED Let$\epsilon,\delta>0$and$n$be sufficiently large in terms of$\epsilon$and$\delta$. Let$G$be a triangle-free graph on$n$vertices with maximum degree$<n^{1/2-\epsilon}$. Can$G$be made into a triangle-free graph with diameter$2$by adding at most$\delta n^2$edges? Asked by Erdős and Gyárfás, who proved that this is the case when$G$has maximum degree$\ll \log n/\log\log n$. A construction of Simonovits shows that this conjecture is false if we just have maximum degree$\leq Cn^{1/2}$, for some large enough$C$. In this note Alon solves this problem in a strong form, in particular proving that a triangle-free graph on$n$vertices with maximum degree$<n^{1/2-\epsilon}$can be made into a triangle-free graph with diameter$2$by adding at most$O(n^{2-\epsilon})$edges. See also [618]. Additional thanks to: Noga Alon OPEN -$250
Let $A\subset \mathbb{R}^2$ be a set of $n$ points such that any subset of size $4$ determines at least $5$ distinct distances. Must $A$ determine $\gg n^2$ many distances?
Erdős also makes the even stronger conjecture that $A$ must contain $\gg n$ many points such that all pairwise distances are distinct.
SOLVED
Let $f(n)$ be the smallest number of colours required to colour the edges of $K_n$ such that every $K_4$ contains at least 5 colours. Determine the size of $f(n)$.
Asked by Erdős and Gyárfás, who proved that $\frac{5}{6}(n-1) < f(n)<n,$ and that $f(9)=8$. Erdős believed the upper bound is closer to the truth. In fact the lower bound is: Bennett, Cushman, Dudek,and Pralat [BCDP22] have shown that $f(n) \sim \frac{5}{6}n.$ Joos and Mubayi [JoMu22] have found a shorter proof of this.
OPEN
Let $k>3$. Does the product of any $k$ consecutive integers $N=\prod_{m< n\leq m+k}n$ (with $m>k$) have a prime factor $p\mid N$ such that $p^2\nmid N$?
Erdős and Selfridge proved that $N$ can never be a perfect power. Erdős remarked that this 'seems hopeless at present'.
OPEN
Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$-coloured there must exist a monochromatic $k$-term arithmetic progression. Improve the bounds for $W(k)$ - for example, prove that $W(k)^{1/k}\to \infty$.
When $p$ is prime Berlekamp [Be68] has proved $W(p+1)\geq p2^p$. Gowers [Go01] has proved $W(k) \leq 2^{2^{2^{2^{2^{k+9}}}}}.$

In [Er81] Erdős further asks whether $W(k+1)/W(k)\to \infty$, or $W(k+1)-W(k)\to \infty$.

SOLVED - $1000 Let$r_k(N)$be the size of the largest subset of$\{1,\ldots,N\}$which does not contain a non-trivial$k$-term arithmetic progression. Prove that$r_k(N)=o(N)$. Proved by Szemerédi [Sz74]. The best known bounds are due to Kelley and Meka [KeMe23] for$k=3$(with further slight improvements in [BlSi23]), Green and Tao [GrTa17] for$k=4$, and Leng, Sah, and Sawhney [LSS24] for$k\geq 5$. See also [3]. Additional thanks to: Zachary Chase SOLVED -$500
Let $r_3(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain a non-trivial $3$-term arithmetic progression. Prove that $r_3(N)\ll N/(\log N)^C$ for every $C>0$.
Proved by Kelley and Meka [KeMe23]. In [ErGr80] and [Er81] it is conjectured that this holds for every $k$-term arithmetic progression.

OPEN
Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression?
Green and Tao [GrTa08] have proved that there must always exist some $k$ primes in arithmetic progression, but these need not be consecutive. Erdős called this conjecture 'completely hopeless at present'.
OPEN - $10000 Let$r_k(N)$be the largest possible size of a subset of$\{1,\ldots,N\}$that does not contain any non-trivial$k$-term arithmetic progression. Prove an asymptotic formula for$r_k(N)$. Erdős remarked this is 'probably unattackable at present'. In [Er97c] Erdős offered \$1000, but given that he elsewhere offered \$5000 just for (essentially) showing that$r_k(N)=o_k(N/\log N)$, that value seems odd. In [Er81] he offers \$10000, stating it is 'probably enormously difficult'.

The best known upper bounds for $r_k(N)$ are due to Kelley and Meka [KeMe23] for $k=3$, Green and Tao [GrTa17] for $k=4$, and Leng, Sah, and Sawhney [LSS24] for $k\geq 5$. An asymptotic formula is still far out of reach, even for $k=3$.

OPEN - $500 Let$A\subset (1,\infty)$be a countably infinite set such that for all$x\neq y\in A$and integers$k\geq 1$we have $\lvert kx -y\rvert \geq 1.$ Does this imply that $\liminf \frac{\lvert A\cap [1,x]\rvert}{x}=0?$ Or $\sum_{x\in A}\frac{1}{x\log x}<\infty,$ or $\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n)?$ Perhaps even $\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}\ll \frac{\log x}{\sqrt{\log\log x}}?$ Note that if$A$is a set of integers then the condition implies that$A$is a primitive set (that is, no element of$A$is divisible by any other), for which the convergence of$\sum_{n\in A}\frac{1}{n\log n}$was proved by Erdős [Er35], and that$\sum_{n<x}\frac{1}{n}=o(\log x)$was proved by Behrend [Be35]. In [Er73] mentions an unpublished proof of Haight that $\lim \frac{\lvert A\cap [1,x]\rvert}{x}=0$ holds if the elements of$A$are independent over$\mathbb{Q}$. Additional thanks to: Zachary Chase SOLVED -$250
The density of integers which have two divisors $d_1,d_2$ such that $d_1<d_2<2d_1$ exists and is equal to $1$.
In [Er79] asks the stronger version with $2$ replaced by any constant $c>1$.

The answer is yes (also to this stronger version), proved by Maier and Tenenbaum [MaTe84]. (Tenenbaum has told me that they received \$650 for their solution.) See also [449]. OPEN Let$s_1<s_2<\cdots$be the sequence of squarefree numbers. Is it true that, for any$\alpha \geq 0$, $\lim_{x\to \infty}\frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha$ exists? Erdős [Er51] proved this for all$0\leq \alpha \leq 2$, and Hooley [Ho73] extended this to all$\alpha \leq 3$. See also [208]. OPEN -$500
If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then $\mathrm{ex}(n;H) \ll n^{2-1/r}.$
Conjectured by Erdős and Simonovits. Open even for $r=3$. Alon, Krivelevich, and Sudakov [AKS03] have proved $\mathrm{ex}(n;H) \ll n^{2-1/4r}.$ They also prove the full Erdős-Simonovits conjectured bound if $H$ is bipartite and the maximum degree in one component is $r$.

SOLVED - $500 If$H$is bipartite and is not$r$-degenerate, that is, there exists an induced subgraph of$H$with minimum degree$>r$then $\mathrm{ex}(n;H) > n^{2-\frac{1}{r}+\epsilon}.$ Conjectured by Erdős and Simonovits. Disproved by Janzer [Ja21], who constructed, for any$\epsilon>0$, a$3$-regular bipartite graph$H$such that $\mathrm{ex}(n;H)\ll n^{\frac{4}{3}+\epsilon}.$ See also [113], [146], and [714]. Additional thanks to: Zachary Hunter OPEN Let$F(k)$be the number of solutions to $1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},$ where$1\leq n_1<\cdots<n_k$are distinct integers. Find good estimates for$F(k)$. The current best bounds known are $2^{c^{\frac{k}{\log k}}}\leq F(k) \leq c_0^{(\frac{1}{5}+o(1))2^k},$ where$c>0$is some absolute constant and$c_0=1.26408\cdots$is the 'Vardi constant'. The lower bound is due to Konyagin [Ko14] and the upper bound to Elsholtz and Planitzer [ElPl21]. OPEN Let$G$be a graph of maximum degree$\Delta$. Is$G$the union of at most$\tfrac{5}{4}\Delta^2$sets of strongly independent edges (sets such that the induced subgraph is the union of vertex-disjoint edges)? Asked by Erdős and Nešetřil. They also asked the easier problem of whether$G$containing at least$\tfrac{5}{4}\Delta^2$many edges implies$G$containing two strongly independent edges. This was proved independently by Chung-Trotter and Gyárfás-Tuza. OPEN A minimal cut of a graph is a minimal set of vertices whose removal disconnects the graph. Let$c(n)$be the maximum number of minimal cuts a graph on$n$vertices can have. Is$c(3m+2)=3^m$? Does$c(n)^{1/n}\to \alpha$for some$\alpha <2$? Asked by Erdős and Nešetřil. Seymour observed that$c(3m+2)\geq 3^m$, as seen by the graph of$m$independent paths of length$4$joining two vertices. OPEN Let$h(n)$be minimal such that, for every graph$G$on$n$vertices, there is a set of vertices$X$of size$\lvert X\rvert\leq h(n)$such that every maximal clique (on at least$2$vertices) in$G$contains at least one vertex from$X$. Let$H(n)$be maximal such that every triangle-free graph on$n$vertices contains an independent set on$H(n)$vertices. Does$h(n)=n-H(n)$? It is easy to see that$h(n)\leq n-\sqrt{n}$and that$h(n)\leq n-H(n)$. Conjectured by Erdős and Gallai, who were unable to make progress even assuming$G$is$K_4$-free. Erdős remarked that this conjecture is 'perhaps completely wrongheaded'. Additional thanks to: Zachary Chase OPEN For any$M\geq 1$, if$A\subset \mathbb{N}$is a sufficiently large finite Sidon set then there are at least$M$many$a\in A+A$such that$a+1,a-1\not\in A+A$. There may even be$\gg \lvert A\rvert^2$many such$a$. A similar question can be asked for truncations of infinite Sidon sets. Additional thanks to: Cedric Pilatte OPEN Let$A$be a finite Sidon set and$A+A=\{s_1<\cdots<s_t\}$. Is it true that $\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty$ as$\lvert A\rvert\to \infty$? A similar problem can be asked for infinite Sidon sets. OPEN Let$A\subset \{1,\ldots,N\}$be a Sidon set with$\lvert A\rvert\sim N^{1/2}$. Must$A+A$be well-distributed over all small moduli? In particular, must about half the elements of$A+A$be even and half odd? Lindström [Li98] has shown this is true for$A$itself, subsequently strengthened by Kolountzakis [Ko99]. OPEN Let$F(N)$be the size of the largest Sidon subset of$\{1,\ldots,N\}$. Is it true that for every$k\geq 1$we have $F(N+k)\leq F(N)+1$ for all sufficiently large$N$? This may even hold with$k\approx \epsilon N^{1/2}$. OPEN Does there exist a maximal Sidon set$A\subset \{1,\ldots,N\}$of size$O(N^{1/3})$? A question of Erdős, Sárközy, and Sós [ESS94]. It is easy to prove that a maximal Sidon set in$\{1,\ldots,N\}$has size$\gg N^{1/3}$. Ruzsa [Ru98b] constructed a maximal Sidon set of size$\ll (N\log N)^{1/3}$. SOLVED Does there exist an infinite Sidon set which is an asymptotic basis of order 3? Yes, as shown by Pilatte [Pi23]. OPEN Let$A\subset \mathbb{N}$be an infinite set such that, for any$n$, there are most$2$solutions to$a+b=n$with$a\leq b$. Must $\liminf_{N\to\infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0?$ If we replace$2$by$1$then$A$is a Sidon set, for which Erdős proved this is true. OPEN There exists some constant$c>0$such that $$R(C_4,K_n) \ll n^{2-c}.$$ The current bounds are $\frac{n^{3/2}}{(\log n)^{3/2}}\ll R(C_4,K_n)\ll \frac{n^2}{(\log n)^2}.$ The upper bound is due to Szemerédi (mentioned in [EFRS78]), and the lower bound is due to Spencer [Sp77]. This problem is #17 in Ramsey Theory in the graphs problem collection. OPEN Let$h(N)$be the smallest$k$such that$\{1,\ldots,N\}$can be coloured with$k$colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate$h(N)$. Investigated by Erdős and Freud. This has been discussed on MathOverflow, where LeechLattice shows $h(N) \ll N^{2/3}.$ The observation of Zachary Hunter in that question coupled with the bounds of Kelley-Meka [KeMe23] imply that $h(N) \gg \exp(c(\log N)^{1/12})$ for some$c>0$. Additional thanks to: Zachary Hunter OPEN -$500
Let $\alpha\in[0,1/2)$ and $n,t\geq 1$. Let $F^{(t)}(n,\alpha)$ be the largest $m$ such that we can $2$-colour the edges of the complete $t$-uniform hypergraph on $n$ vertices such that if $X\subseteq [n]$ with $\lvert X\rvert \geq m$ then there are at least $\alpha \binom{\lvert X\rvert}{t}$ many $t$-subsets of $X$ of each colour.

For fixed $n,t$ as we change $\alpha$ from $0$ to $1/2$ does $F^{(t)}(n,\alpha)$ increase continuously or are there jumps? Only one jump?

For $\alpha=0$ this is the usual Ramsey function. A conjecture of Erdős, Hajnal, and Rado (see [562]) implies that $F^{(t)}(n,0)\asymp \log_{t-1} n$ and results of Erdős and Spencer imply that $F^{(t)}(n,\alpha) \asymp_\alpha (\log n)^{\frac{1}{t-1}}$ for $\alpha$ close to $1/2$.

Erdős believed there might be just one jump, occcurring at $\alpha=0$.

OPEN
Let $\alpha>0$ and $n\geq 1$. Let $F(n,\alpha)$ be the largest $k$ such that in any 2-colouring of the edges of $K_n$ any subgraph $H$ on at least $k$ vertices contains more than $\alpha\binom{\lvert H\rvert}{2}$ many edges of each colour.

Prove that for every fixed $0\leq \alpha \leq 1/2$, as $n\to\infty$, $F(n,\alpha)\sim c_\alpha \log n$ for some constant $c_\alpha$.

It is easy to show with the probabilistic method that there exist $c_1(\alpha),c_2(\alpha)$ such that $c_1(\alpha)\log n < F(n,\alpha) < c_2(\alpha)\log n.$
SOLVED
For any $d\geq 1$ if $H$ is a graph such that every subgraph contains a vertex of degree at most $d$ then $R(H)\ll_d n$.
The Burr-Erdős conjecture. This is equivalent to showing that if $H$ is the union of $c$ forests then $R(H)\ll_c n$, and also that if every subgraph has average degree at most $d$ then $R(H)\ll_d n$. Solved by Lee [Le16], who proved that $R(H) \leq 2^{2^{O(d)}}n.$

This problem is #9 in Ramsey Theory in the graphs problem collection. See also [800].

SOLVED
A set $A\subset \mathbb{N}$ is primitive if no member of $A$ divides another. Is the sum $\sum_{n\in A}\frac{1}{n\log n}$ maximised over all primitive sets when $A$ is the set of primes?
Erdős [Er35] proved that this sum always converges for a primitive set. Solved by Lichtman [Li23].
OPEN - $250 Give an asymptotic formula for$R(3,k)$. It is known that there exists some constant$c>0$such that for large$k$$c\frac{k^2}{\log k}\leq R(3,k) \leq (1+o(1))\frac{k^2}{\log k}.$ The lower bound is due to Kim [Ki95], the upper bound is due to Shearer [Sh83], improving an earlier bound of Ajtai, Komlós, and Szemerédi [AjKoSz80]. The lower bound has been improved to $\left(\frac{1}{4}-o(1)\right)\frac{k^2}{\log k}$ independently by Bohman and Keevash [BoKe21] and Pontiveros, Griffiths and Morris [PGM20]. The latter collection of authors conjecture that this lower bound is the true order of magnitude. See also [544]. SOLVED -$250
Prove that $R(4,k) \gg \frac{k^3}{(\log k)^{O(1)}}.$
Spencer [Sp77] proved $R(4,k) \gg (k\log k)^{5/2}.$ Ajtai, Komlós, and Szemerédi [AKS80] proved $R(4,k) \ll \frac{k^3}{(\log k)^2}.$ This is true, and was proved by Mattheus and Verstraete [MaVe23], who showed that $R(4,k) \gg \frac{k^3}{(\log k)^4}.$

This problem is #5 in Ramsey Theory in the graphs problem collection.

OPEN
If $G$ is a graph with at most $k$ edge disjoint triangles then can $G$ be made triangle-free after removing at most $2k$ edges?
A problem of Tuza. It is trivial that $G$ can be made triangle-free after removing at most $3k$ edges. The examples of $K_4$ and $K_5$ show that $2k$ would be best possible.
OPEN
Let $F(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain any set of the form $\{n,2n,3n\}$. What is $\lim_{N\to \infty}\frac{F(N)}{N}?$ Is this limit irrational?
This limit was proved to exist by Graham, Spencer, and Witsenhausen [GrSpWi77]. Similar questions can be asked for the density or upper density of infinite sets without such configurations.
OPEN
Let $k\geq 3$ and $f(k)$ be the supremum of $\sum_{n\in A}\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-term arithmetic progression. Estimate $f(k)$.

Is $\lim_{k\to \infty}\frac{f(k)}{\log W(k)}=\infty$ where $W(k)$ is the van der Waerden number?

Gerver [Ge77] has proved $f(k) \geq (1+o(1))k\log k.$ It is trivial that $\frac{f(k)}{\log W(k)}\geq \frac{1}{2},$ but improving the right-hand side to any constant $>1/2$ is open.
OPEN
Let $F(N)$ be the smallest possible size of $A\subset \{0,1,\ldots,N\}$ such that $\{0,1,\ldots,N\}\subset A-A$. Find the value of $\lim_{N\to \infty}\frac{F(N)}{N^{1/2}}.$
The Sparse Ruler problem. Rédei asked whether this limit exists, which was proved by Erdős and Gál [ErGa48]. Bounds on the limit were improved by Leech [Le56]. The limit is known to be in the interval $[1.56,\sqrt{3}]$. The lower bound is due to Leech [Le56], the upper bound is due to Wichmann [Wi63]. Computational evidence by Pegg [Pe20] suggests that the upper bound is the truth. A similar question can be asked without the restriction $A\subset \{0,1,\ldots,N\}$.
SOLVED
Is it true that for every $\epsilon>0$ and integer $t\geq 1$, if $N$ is sufficiently large and $A$ is a subset of $[t]^N$ of size at least $\epsilon t^N$ then $A$ must contain a combinatorial line $P$ (a set $P=\{p_1,\ldots,p_t\}$ where for each coordinate $1\leq j\leq t$ the $j$th coordinate of $p_i$ is either $i$ or constant).
The 'density Hales-Jewett' problem. This was proved by Furstenberg and Katznelson [FuKa91]. A new elementary proof, which gives quantitative bounds, was proved by the Polymath project [Po09].
OPEN
Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?
First asked by Hindman. Hindman [Hi80] has proved this is false (with 7 colours) if we ask for an infinite $A$.

Moreira [Mo17] has proved that in any finite colouring of $\mathbb{N}$ there exist $x,y$ such that $\{x,x+y,xy\}$ are all the same colour.

Alweiss [Al23] has proved that, in any finite colouring of $\mathbb{Q}\backslash \{0\}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour. Bowen and Sabok [BoSa22] had proved this earlier for the first non-trivial case of $\lvert A\rvert=2$.

OPEN
In any $2$-colouring of $\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$.
For some colourings a single equilateral triangle has to be excluded, considering the colouring by alternating strips. Shader [Sh76] has proved this is true if we just consider a single right-angled triangle.
OPEN
A finite set $A\subset \mathbb{R}^n$ is called Ramsey if, for any $k\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring of $\mathbb{R}^d$ there exists a monochromatic copy of $A$. Characterise the Ramsey sets in $\mathbb{R}^n$.
Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus [EGMRSS73] proved that every Ramsey set is 'spherical': it lies on the surface of some sphere. Graham has conjectured that every spherical set is Ramsey. Leader, Russell, and Walters [LRW12] have alternatively conjectured that a set is Ramsey if and only if it is 'subtransitive': it can be embedded in some higher-dimensional set on which rotations act transitively.

Sets known to be Ramsey include vertices of $k$-dimensional rectangles [EGMRSS73], non-degenerate simplices [FrRo90], trapezoids [Kr92], and regular polygons/polyhedra [Kr91].

SOLVED
Show that, for any $n\geq 5$, the binomial coefficient $\binom{2n}{n}$ is not squarefree.
It is easy to see that $4\mid \binom{2n}{n}$ except when $n=2^k$, and hence it suffices to prove this when $n$ is a power of $2$.

Proved by Sárkzözy [Sa85] for all sufficiently large $n$, and by Granville and Ramaré [GrRa96] for all $n\geq 5$.

More generally, if $f(n)$ is the largest integer such that, for some prime $p$, we have $p^{f(n)}$ dividing $\binom{2n}{n}$, then $f(n)$ should tend to infinity with $n$. Can one even disprove that $f(n)\gg \log n$?

OPEN
Let $N(k,\ell)$ be the minimal $N$ such that for any $f:\{1,\ldots,N\}\to\{-1,1\}$ there must exist a $k$-term arithmetic progression $P$ such that $\left\lvert \sum_{n\in P}f(n)\right\rvert> \ell.$ Find good upper bounds for $N(k,\ell)$. Is it true that for any $c>0$ there exists some $C>1$ such that $N(k,ck)\leq C^k?$ What about $N(k,1)\leq C^k$ or $N(k,\sqrt{k})\leq C^k?$
No decent bound is known even for $N(k,1)$. Probabilistic methods imply that, for every fixed constant $c>0$, we have $N(k,ck)>C_c^k$ for some $C_c>1$.
OPEN
Find the smallest $h(d)$ such that the following holds. There exists a function $f:\mathbb{N}\to\{-1,1\}$ such that, for every $d\geq 1$, $\max_{P_d}\left\lvert \sum_{n\in P_d}f(n)\right\rvert\leq h(d),$ where $P_d$ ranges over all finite arithmetic progressions with common difference $d$.
Cantor, Erdős, Schreiber, and Straus [Er66] proved that $h(d)\ll d!$ is possible. Van der Waerden's theorem implies that $h(d)\to \infty$. Beck [Be17] has shown that $h(d) \leq d^{8+\epsilon}$ is possible for every $\epsilon>0$. Roth's famous discrepancy lower bound [Ro64] implies that $h(d)\gg d^{1/2}$.
SOLVED
Let $A_1,A_2,\ldots$ be an infinite collection of infinite sets of integers, say $A_i=\{a_{i1}<a_{i2}<\cdots\}$. Does there exist some $f:\mathbb{N}\to\{-1,1\}$ such that $\max_{m, 1\leq i\leq d} \left\lvert \sum_{1\leq j\leq m} f(a_{ij})\right\rvert \ll_d 1$ for all $d\geq 1$?
Erdős remarks 'it seems certain that the answer is affirmative'. This was solved by Beck [Be81]. Recently Beck [Be17] proved that one can replace $\ll_d 1$ with $\ll d^{4+\epsilon}$ for any $\epsilon>0$.
SOLVED
Let $1\leq k<\ell$ be integers and define $F_k(N,\ell)$ to be minimal such that every set $A\subset \mathbb{N}$ of size $N$ which contains at least $F_k(N,\ell)$ many $k$-term arithmetic progressions must contain an $\ell$-term arithmetic progression. Find good upper bounds for $F_k(N,\ell)$. Is it true that $F_3(N,4)=o(N^2)?$ Is it true that for every $\ell>3$ $\lim_{N\to \infty}\frac{\log F_3(N,\ell)}{\log N}=2?$
Erdős remarks the upper bound $o(N^2)$ is certainly false for $\ell >\epsilon \log N$. The answer is yes: Fox and Pohoata [FoPo20] have shown that, for all fixed $1\leq k<\ell$, $F_k(N,\ell)=N^{2-o(1)}$ and in fact $F_{k}(N,\ell) \leq \frac{N^2}{(\log\log N)^{C_\ell}}$ where $C_\ell>0$ is some constant.
OPEN
If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\mathcal{F}$. Note that it is trivial that $\mathrm{ex}(n;\mathcal{F})\leq \mathrm{ex}(n;G)$ for every $G\in\mathcal{F}$.

Is it true that, for every $\mathcal{F}$, there exists $G\in\mathcal{F}$ such that $\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?$

A problem of Erdős and Simonovits.

This is trivially true if $\mathcal{F}$ does not contain any bipartite graphs, since by the Erdős-Stone theorem if $H\in\mathcal{F}$ has minimal chromatic number $r\geq 2$ then $\mathrm{ex}(n;H)=\mathrm{ex}(n;\mathcal{F})=\left(\frac{r-2}{r-1}+o(1)\right)\binom{n}{2}.$ Erdős and Simonovits observe that this is false for infinite families $\mathcal{F}$, e.g. the family of all cycles.

OPEN
Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that $R(Q_n) \ll 2^n.$
Conjectured by Burr and Erdős. The trivial bound is $R(Q_n) \leq R(K_{2^n})\leq C^{2^n}$ for some constant $C>1$. This was improved a number of times; the current best bound due to Tikhomirov [Ti22] is $R(Q_n)\ll 2^{(2-c)n}$ for some small constant $c>0$. (In fact $c\approx 0.03656$ is permissible.)
SOLVED
Let $k\geq 3$. What is the maximum number of edges that a graph on $n$ vertices can contain if it does not have a $k$-regular subgraph? Is it $\ll n^{1+o(1)}$?
Asked by Erdős and Sauer. Resolved by Janzer and Sudakov [JaSu22], who proved that there exists some $C=C(k)>0$ such that any graph on $n$ vertices with at least $Cn\log\log n$ edges contains a $k$-regular subgraph.

A construction due to Pyber, Rödl, and Szemerédi [PRS95] shows that this is best possible.

OPEN - $250 Let$R(3;k)$be the minimal$n$such that if the edges of$K_n$are coloured with$k$colours then there must exist a monochromatic triangle. Determine $\lim_{k\to \infty}R(3;k)^{1/k}.$ Erdős offers \$100 for showing that this limit is finite. An easy pigeonhole argument shows that $R(3;k)\leq 2+k(R(3;k-1)-1),$ from which $R(3;k)\leq \lceil e k!\rceil$ immediately follows. The best-known upper bounds are all of the form $ck!+O(1)$, and arise from this type of inductive relationship and computational bounds for $R(3;k)$ for small $k$. The best-known lower bound (coming from lower bounds for Schur numbers) is due to Exoo [Ex94], $R(3;k) \gg (321)^{k/5}.$

Additional thanks to: Antonio Girao, David Penman
OPEN
Any graph on $n$ vertices can be decomposed into $O(n)$ many cycles and edges.
Conjectured by Erdős and Gallai, who proved that $O(n\log n)$ many cycles and edges suffices.

The best bound available is due to Bucić and Montgomery [BM22], who prove that $O(n\log^*n)$ many cycles and edges suffice, where $\log^*$ is the iterated logarithm function.

SOLVED
Let $f_3(n)$ be the maximal size of a subset of $\{0,1,2\}^n$ which contains no three points on a line. Is it true that $f_3(n)=o(3^n)$?
Originally considered by Moser. It is trivial that $f_3(n)\geq R_3(3^n)$, the maximal size of a subset of $\{1,\ldots,3^n\}$ without a three-term arithmetic progression. Moser showed that $f_3(n) \gg \frac{3^n}{\sqrt{n}}.$

The answer is yes, which is a corollary of the density Hales-Jewett theorem, proved by Furstenberg and Katznelson [FuKa91].

OPEN
Let $F(N)$ be the maximal size of $A\subseteq \{1,\ldots,N\}$ which is 'non-averaging', so that no $n\in A$ is the arithmetic mean of at least two elements in $A$. What is the order of growth of $F(N)$?
Originally due to Straus. It is known that $N^{1/4}\ll F(N) \ll N^{\sqrt{2}-1+o(1)}.$ The lower bound is due to Bosznay [Bo89] and the upper bound to Conlon, Fox, and Pham [CFP23] (improving on earlier bound due to Erdős and Sárközy [ErSa90] of $\ll (N\log N)^{1/2}$).

OPEN
Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference $d$ of length $f(d)$ for infinitely many $d$.
Originally asked by Cohen. Erdős observed that colouring according to whether $\{ \sqrt{2}n\}<1/2$ or not implies $f(d) \ll d$ (using the fact that $\|\sqrt{2}q\| \gg 1/q$ for all $q$, where $\|x\|$ is the distance to the nearest integer). Beck [Be80] has improved this using the probabilistic method, constructing a colouring that shows $f(d)\leq (1+o(1))\log_2 d$. Van der Waerden's theorem implies $f(d)\to \infty$ is necessary.
OPEN
What is the smallest $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points?
Juhász [Ju79] has shown that $k\geq 5$. It is also known that $k\leq 10000000$ (as Erdős writes, 'more or less').
SOLVED
If $\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area?
Graham [Gr80] has shown that this is true if we replace rectangle by right-angled triangle. The same question can be asked for parallelograms. It is not true for rhombuses.

This is false; Kovač [Ko23] provides an explicit (and elegantly simple) colouring using 25 colours such that no colour class contains the vertices of a rectangle of area $1$. The question for parallelograms remains open.

Additional thanks to: Ryan Alweiss, Vjekoslav Kovac
OPEN
Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\{1,\ldots,N\}$ (into any number of colours) there is always either a monochromatic $k$-term arithmetic progression or a rainbow arithmetic progression (i.e. all elements are different colours). Estimate $H(k)$. Is it true that $H(k)^{1/k}/k \to \infty$ as $k\to\infty$?
This type of problem belongs to 'canonical' Ramsey theory. The existence of $H(k)$ follows from Szemerédi's theorem, and it is easy to show that $H(k)^{1/k}\to\infty$.
SOLVED
Let $C>0$ be arbitrary. Is it true that, if $n$ is sufficiently large depending on $C$, then in any $2$-colouring of $\binom{\{2,\ldots,n\}}{2}$ there exists some $X\subset \{2,\ldots,n\}$ such that $\binom{X}{2}$ is monochromatic and $\sum_{x\in X}\frac{1}{\log x}\geq C?$
The answer is yes, which was proved by Rödl [Ro03].

In the same article Rödl also proved a lower bound for this problem, constructing, for all $n$, a $2$-colouring of $\binom{\{2,\ldots,n\}}{2}$ such that if $X\subseteq \{2,\ldots,n\}$ is such that $\binom{X}{2}$ is monochromatic then $\sum_{x\in X}\frac{1}{\log x}\ll \log\log\log n.$

This bound is best possible, as proved by Conlon, Fox, and Sudakov [CFS13], who proved that, if $n$ is sufficiently large, then in any $2$-colouring of $\binom{\{2,\ldots,n\}}{2}$ there exists some $X\subset \{2,\ldots,n\}$ such that $\binom{X}{2}$ is monochromatic and $\sum_{x\in X}\frac{1}{\log x}\geq 2^{-8}\log\log\log n.$

SOLVED
Let $A=\{a_1,a_2,\ldots\}\subset \mathbb{R}^d$ be an infinite sequence such that $a_{i+1}-a_i$ is a positive unit vector (i.e. is of the form $(0,0,\ldots,1,0,\ldots,0)$). For which $d$ must $A$ contain a three-term arithmetic progression?
This is true for $d\leq 3$ and false for $d\geq 4$.

This problem is equivalent to one on 'abelian squares' (see [231]). In particular $A$ can be interpreted as an infinite string over an alphabet with $d$ letters (each letter describining which of the $d$ possible steps is taken at each point). An abelian square in a string $s$ is a pair of consecutive blocks $x$ and $y$ appearing in $s$ such that $y$ is a permutation of $x$. The connection comes from the observation that $p,q,r\in A\subset \mathbb{R}^d$ form a three-term arithmetic progression if and only if the string corresponding to the steps from $p$ to $q$ is a permutation of the string corresponding to the steps from $q$ to $r$.

This problem is therefore equivalent to asking for which $d$ there exists an infinite string over $\{1,\ldots,d\}$ with no abelian squares. It is easy to check that in fact any finite string of length $7$ over $\{1,2,3\}$ contains an abelian square.

An infinite string without abelian squares was constructed when $d=4$ by Keränen [Ke92]. We refer to a recent survey by Fici and Puzynina [FiPu23] for more background and related results, and a blog post by Renan for an entertaining and educational discussion.

Additional thanks to: Boris Alexeev and Dustin Mixon
OPEN
Let $S\subseteq \mathbb{Z}^3$ be a finite set and let $A=\{a_1,a_2,\ldots,\}\subset \mathbb{Z}^3$ be an infinite $S$-walk, so that $a_{i+1}-a_i\in S$ for all $i$. Must $A$ contain three collinear points?
Originally conjectured by Gerver and Ramsey [GeRa79], who showed that the answer is yes for $\mathbb{Z}^2$, and for $\mathbb{Z}^3$ that the largest number of collinear points can be bounded.
SOLVED
Let $k\geq 3$. Must any ordering of $\mathbb{R}$ contain a monotone $k$-term arithmetic progression, that is, some $x_1<\cdots<x_k$ which forms an increasing or decreasing $k$-term arithmetic progression?
The answer is no, even for $k=3$, as shown by Ardal, Brown, and Jungić [ABJ11].

OPEN
What is the smallest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\cdots<x_k$?
Geneson [Ge19] proved that $k\leq 5$. Adenwalla [Ad22] proved that $k\leq 4$.

Additional thanks to: Boris Alexeev and Dustin Mixon
OPEN
Must every permutation of $\mathbb{N}$ contain a monotone 4-term arithmetic progression $x_1<x_2<x_3<x_4$?
Davis, Entringer, Graham, and Simmons [DEGS77] have shown that there must exist a monotone 3-term arithmetic progression and need not contain a 5-term arithmetic progression.

Additional thanks to: Boris Alexeev and Dustin Mixon
OPEN
Can $\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?
If three sets are allowed then this is possible.
Additional thanks to: Boris Alexeev and Dustin Mixon
SOLVED
If $A\subset \mathbb{N}$ is a Sidon set then must the complement of $A$ contain an infinite arithmetic progression?
The answer is yes, as shown by Baumgartner [Ba75].
SOLVED
If $A\subset \mathbb{R}$ does not contain a 3-term arithmetic progression then must $\mathbb{R}\backslash A$ contain an infinite arithmetic progression?
The answer is no, as shown by Baumgartner [Ba75] (whose construction uses the axiom of choice).
OPEN
Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$?
It follows from the prime number theorem that such a progression has length $\leq(1+o(1))\log N$.
OPEN
Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\{1,\ldots,N\}$ without a $k$-term arithmetic progression? Is it true that $\lim_{N\to \infty}\frac{R_3(N)}{G_3(N)}=1?$
First asked and investigated by Riddell [Ri69]. It is trivial that $G_k(N)\leq R_k(N)$, and it is possible that $G_k(N) <R_k(N)$ (for example with $k=3$ and $N=14$). Komlós, Sulyok, and Szemerédi [KSS75] have shown that $R_k(N) \ll_k G_k(N)$.
OPEN
Let $n_1<\cdots < n_r\leq N$ with associated $a_i\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\equiv a_i\pmod{n_i}$ for at most one $1\leq i\leq r$). How large can $r$ be in terms of $N$?
Let $f(N)$ be the maximum possible $r$. Erdős and Stein conjectured that $f(N)=o(N)$, which was proved by Erdős and Szemerédi [ErSz68], who showed that, for every $\epsilon>0$, $\frac{N}{\exp((\log N)^{1/2+\epsilon})} \ll_\epsilon f(N) < \frac{N}{(\log N)^c}$ for some $c>0$. Erdős believed the lower bound is closer to the truth.
OPEN
Is there an integer $m$ with $(m,6)=1$ such that none of $2^k3^\ell m+1$ are prime, for any $k,\ell\geq 0$?
There are odd integers $m$ such that none of $2^km+1$ are prime, which arise from covering systems (i.e. one shows that there exists some $n$ such that $(2^km+1,n)>1$ for all $k\geq 1$). Erdős and Graham also ask whether there is such an $m$ where a covering system is not 'responsible'. The answer is probably no since otherwise this would imply there are infinitely many Fermat primes of the form $2^{2^t}+1$.
OPEN
Do there exist $n$ with an associated covering system $a_d\pmod{d}$ on the divisors of $n$ (so that $d\mid n$ for all $d$ in the system), such that if $x\equiv a_d\pmod{d}\textrm{ and }x\equiv a_{d'}\pmod{d'}$ then $(d,d')=1$? For a given $n$ what is the density of the integers which do not satisfy any of the congruences?
The density of such $n$ is zero. Erdős and Graham believed that no such $n$ exist.
OPEN
Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$? (Here $\Omega(m)$ is the number of prime divisors of $m$ counted with multiplicity.) What about $<\epsilon \log\log m$? Or some more slowly growing function?
It is easy to see by probabilistic methods that this holds for almost all integers. Romanoff [Ro34] showed that a positive density set of integers are representable as the sum of $2^k+p$ for prime $p$, and Erdős used covering systems to show that there is a positive density set of integers which cannot be so represented.
SOLVED
Let $x>0$ be a real number. For any $n\geq 1$ let $R_n(x) = \sum_{i=1}^n\frac{1}{m_i}<x$ be the maximal sum of $n$ distinct unit fractions which is $<x$.

Is it true that, for almost all $x$, for sufficiently large $n$, we have $R_{n+1}(x)=R_n(x)+\frac{1}{m},$ where $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is $<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?)

Erdős and Graham write it is 'not difficult' to construct irrational $x$ such that this fails (although give no proof or reference, and it seems to still be an open problem to actually construct some such irrational $x$). Curtiss [Cu22] showed that this is true for $x=1$ and Erdős [Er50b] showed it is true for all $x=1/m$ with $m\geq 1$. Nathanson [Na23] has shown it is true for $x=a/b$ when $a\mid b+1$ and Chu [Ch23b] has shown it is true for a larger class of rationals; it is still unknown whether this is true for all rational $x>0$.

Without the 'eventually' condition this can fail for some rational $x$ (although Erdős [Er50b] showed it holds without the eventually for rationals of the form $1/m$). For example $R_1(\tfrac{11}{24})=\frac{1}{3}$ but $R_2(\tfrac{11}{24})=\frac{1}{4}+\frac{1}{5}.$

Kovač [Ko24b] has proved that this is false - in fact as false as possible: the set of $x\in (0,\infty)$ for which the best underapproximations are eventually 'greedy' has Lebesgue measure zero. (It remains an open problem to give any explicit example of such a number.)

SOLVED
For any $g\geq 2$, if $n$ is sufficiently large and $\equiv 1,3\pmod{6}$ then there exists a 3-uniform hypergraph on $n$ vertices such that
• every pair of vertices is contained in exactly one edge (i.e. the graph is a Steiner triple system) and
• for any $2\leq j\leq g$ any collection of $j$ edges contains at least $j+3$ vertices.
Proved by Kwan, Sah, Sawhney, and Simkin [KSSS22b].
OPEN
Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\epsilon>0$ and large $n$, $s_{n+1}-s_n \ll_\epsilon s_n^{\epsilon}?$ Is it true that $s_{n+1}-s_n \leq (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}?$
Erdős [Er51] showed that there are infinitely many $n$ such that $s_{n+1}-s_n > (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n},$ so this bound would be the best possible.

In [Er79] Erdős says perhaps $s_{n+1}-s_n \ll \log s_n$, but he is 'very doubtful'.

Filaseta and Trifonov [FiTr92] proved an upper bound of $s_n^{1/5}$. Pandey [Pa24] has improved this exponent to $1/5-c$ for some constant $c>0$.

SOLVED
Let $A$ be a finite collection of $d\geq 4$ non-parallel lines in $\mathbb{R}^2$ such that there are no points where at least four lines from $A$ meet. Must there exist a 'Gallai triangle' (or 'ordinary triangle'): three lines from $A$ which intersect in three points, and each of these intersection points only intersects two lines from $A$?
Equivalently, one can ask the dual problem: given $n$ points in $\mathbb{R}^2$ such that there are no lines containing at least four points then there are three points such that the lines determined by them are ordinary ones (i.e. contain exactly two points each).

The Sylvester-Gallai theorem implies that there must exist a point where only two lines from $A$ meet. This problem asks whether there must exist three such points which form a triangle (with sides induced by lines from $A$). Füredi and Palásti [FuPa84] showed this is false when $d\geq 4$ is not divisible by $9$. Escudero [Es16] showed this is false for all $d\geq 4$.

SOLVED
Let $f(n)$ be minimal such that the following holds. For any $n$ points in $\mathbb{R}^2$, not all on a line, there must be at least $f(n)$ many lines which contain exactly 2 points (called 'ordinary lines'). Does $f(n)\to \infty$? How fast?
Conjectured by Erdős and de Bruijn. The Sylvester-Gallai theorem states that $f(n)\geq 1$. The fact that $f(n)\geq 1$ was conjectured by Sylvester in 1893. Erdős rediscovered this conjecture in 1933 and told it to Gallai who proved it.

That $f(n)\to \infty$ was proved by Motzkin [Mo51]. Kelly and Moser [KeMo58] proved that $f(n)\geq\tfrac{3}{7}n$ for all $n$. This is best possible for $n=7$. Motzkin conjectured that for $n\geq 13$ there are at least $n/2$ such lines. Csima and Sawyer [CsSa93] proved a lower bound of $f(n)\geq \tfrac{6}{13}n$ when $n\geq 8$. Green and Tao [GrTa13] proved that $f(n)\geq n/2$ for sufficiently large $n$. (A proof that $f(n)\geq n/2$ for large $n$ was earlier claimed by Hansen but this proof was flawed.)

The bound of $n/2$ is best possible for even $n$, since one could take $n/2$ points on a circle and $n/2$ points at infinity. Surprisingly, Green and Tao [GrTa13] show that if $n$ is odd then $f(n)\geq 3\lfloor n/4\rfloor$.

SOLVED - $100 Let$1\leq k<n$. Given$n$points in$\mathbb{R}^2$, at most$n-k$on any line, there are$\gg kn$many lines which contain at least two points. In particular, given any$2n$points with at most$n$on a line there are$\gg n^2$many lines formed by the points. Solved by Beck [Be83] and Szemerédi and Trotter [SzTr83]. In [Er84] Erdős speculates that perhaps there are$\geq (1+o(1))kn/6$many such lines, but says 'perhaps [this] is too optimistic and one should first look for a counterexample'. The constant$1/6$would be best possible here, since there are arrangements of$n$points with no four points on a line and$\sim n^2/6$many lines containing three points (see Burr, Grünbaum, and Sloane [BGS74] and Füredi and Palásti [FuPa84]). OPEN Is there a dense subset of$\mathbb{R}^2$such that all pairwise distances are rational? Conjectured by Ulam. Erdős believed there cannot be such a set. This problem is discussed in a blogpost by Terence Tao, in which he shows that there cannot be such a set, assuming the Bombieri-Lang conjecture. The same conclusion was independently obtained by Shaffaf [Sh18]. Indeed, Shaffaf and Tao actually proved that such a rational distance set must be contained in a finite union of real algebraic curves. Solymosi and de Zeeuw [SdZ10] then proved (unconditionally) that a rational distance set contained in a real algebraic curve must be finite, unless the curve contains a line or a circle. Ascher, Braune, and Turchet [ABT20] observed that, combined, these facts imply that a rational distance set in general position must be finite (conditional on the Bombieri-Lang conjecture). OPEN Let$n\geq 4$. Are there$n$points in$\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers? Anning and Erdős [AnEr45] proved there cannot exist an infinite such set. Harborth constructed such a set when$n=5$. The best construction to date, due to Kreisel and Kurz [KK08], has$n=7$. Ascher, Braune, and Turchet [ABT20] have shown that there is a uniform upper bound on the size of such a set, conditional on the Bombieri-Lang conjecture. Greenfeld, Iliopoulou, and Peluse [GIP24] have shown (unconditionally) that any such set must be very sparse, in that if$S\subseteq [-N,N]^2$has no three on a line and no four on a circle, and all pairwise distances integers, then $\lvert S\rvert \ll (\log N)^{O(1)}.$ See also [130]. SOLVED Let$S\subset \mathbb{R}^2$be such that no two points in$S$are distance$1$apart. Must the complement of$S$contain four points which form a unit square? The answer is yes, proved by Juhász [Ju79], who proved more generally that the complement of$S$must contain a congruent copy of any set of four points. This is not true for arbitrarily large sets of points, but perhaps is still true for any set of five points. Additional thanks to: Bhavik Mehta SOLVED Does there exist$S\subseteq \mathbb{R}^2$such that every set congruent to$S$(that is,$S$after some translation and rotation) contains exactly one point from$\mathbb{Z}^2$? An old question of Steinhaus. Erdős was 'almost certain that such a set does not exist'. In fact, such a set does exist, as proved by Jackson and Mauldin [JaMa02]. Their construction depends on the axiom of choice. Additional thanks to: Vjekoslav Kovac SOLVED Let$g(k)$be the smallest integer (if any such exists) such that any$g(k)$points in$\mathbb{R}^2$contains an empty convex$k$-gon (i.e. with no point in the interior). Does$g(k)$exist? If so, estimate$g(k)$. A variant of the 'happy ending' problem [107], which asks for the same without the 'no point in the interior' restriction. Erdős observed$g(4)=5$(as with the happy ending problem) but Harborth [Ha78] showed$g(5)=10$. Nicolás [Ni07] and Gerken [Ge08] independently showed that$g(6)$exists. Horton [Ho83] showed that$g(n)$does not exist for$n\geq 7$. This problem is #2 in Ramsey Theory in the graphs problem collection. Additional thanks to: Zachary Hunter OPEN For which$n$are there$n$points in$\mathbb{R}^2$, no three on a line and no four on a circle, which determine$n-1$distinct distances and so that (in some ordering of the distances) the$i$th distance occurs$i$times? An example with$n=4$is an isosceles triangle with the point in the centre. Erdős originally believed this was impossible for$n\geq 5$, but Pomerance constructed a set with$n=5$(see [Er83c] for a description), and Palásti has proved such sets exist for all$n\leq 8$. Erdős believes this is impossible for all sufficiently large$n$. OPEN Let$d_n=p_{n+1}-p_n$. The set of$n$such that$d_{n+1}\geq d_n$has density$1/2$, and similarly for$d_{n+1}\leq d_n$. Furthermore, there are infinitely many$n$such that$d_{n+1}=d_n$. SOLVED Are there arbitrarily long arithmetic progressions of primes? The answer is yes, proved by Green and Tao [GrTa08]. The stronger claim that there are arbitrarily long arithmetic progressions of consecutive primes is still open. SOLVED -$250
Let $n\geq 1$ and $A=\{a_1<\cdots <a_{\phi(n)}\}=\{ 1\leq m<n : (m,n)=1\}.$ Is it true that $\sum_{1\leq k<\phi(n)}(a_{k+1}-a_k)^2 \ll \frac{n^2}{\phi(n)}?$
The answer is yes, as proved by Montgomery and Vaughan [MoVa86], who in fact found the correct order of magnitude with the power $2$ replaced by any $\gamma\geq 1$ (which was also asked by Erdős in [Er73]).
SOLVED
Is there a set $A\subset\mathbb{N}$ such that, for all large $N$, $\lvert A\cap\{1,\ldots,N\}\rvert \ll N/\log N$ and such that every large integer can be written as $2^k+a$ for some $k\geq 0$ and $a\in A$?
Lorentz [Lo54] proved there is such a set with, for all large $N$, $\lvert A\cap\{1,\ldots,N\}\rvert \ll \frac{\log\log N}{\log N}N$ The answer is yes, proved by Ruzsa [Ru72]. Ruzsa's construction is ingeniously simple: $A = \{ 5^nm : m\geq 1\textrm{ and }5^n\geq C\log m\}+\{0,1\}$ for some large absolute constant $C>0$. That every large integer is of the form $2^k+a$ for some $a\in A$ is a consequence of the fact that $2$ is a primitive root of $5^n$ for all $n\geq 1$.

In [Ru01] Ruzsa constructs an asymptotically best possible answer to this question (a so-called 'exact additive complement'; that is, there is such a set $A$ with $\lvert A\cap\{1,\ldots,N\}\rvert \sim \frac{N}{\log_2N}$ as $N\to \infty$.

OPEN
Let $n_1<n_2<\cdots$ be the sequence of integers which are the sum of two squares. Explore the behaviour of (i.e. find good upper and lower bounds for) the consecutive differences $n_{k+1}-n_k$.
Erdős [Er51] proved that, for infinitely many $k$, $n_{k+1}-n_k \gg \frac{\log n_k}{\sqrt{\log\log n_k}}.$ Richards [Ri82] improved this to $\limsup_{k\to \infty} \frac{n_{k+1}-n_k}{\log n_k} \geq 1/4.$ The constant $1/4$ here has been improved, most lately to $0.868\cdots$ by Dietmann, Elsholtz, Kalmynin, Konyagin, and Maynard [DEKKM22]. The best known upper bound is due to Bambah and Chowla [BaCh47], who proved that $n_{k+1}-n_k \ll n_k^{1/4}.$
OPEN
Let $d\geq 2$ and $n\geq 2$. Let $f_d(n)$ be maximal such that, for any $A\subseteq \mathbb{R}^d$ of size $n$, with diameter $1$, the distance 1 occurs between $f_d(n)$ many pairs of points in $A$. Estimate $f_d(n)$.
Hopf and Pannwitz [HoPa34] proved $f_2(n)=n$. Heppes [He56] and Grünbaum-Strasziewicz independently showed that $f_3(n)=2n-2$.

SOLVED
If $A\subseteq \mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determine an obtuse angle.
For $d=2$ this is trivial. For $d=3$ there is an unpublished proof by Kuiper and Boerdijk. The general case was proved by Danzer and Grünbaum [DaGr62].
Additional thanks to: Boris Alexeev and Dustin Mixon
SOLVED
Let $f(\theta) = \sum_{k\geq 1}c_k e^{ik\theta}$ be a trigonometric polynomial (so that the $c_k\in \mathbb{C}$ are finitely supported) with real roots such that $\max_{\theta\in [0,2\pi]}\lvert f(\theta)\rvert=1$. Then $\int_0^{2\pi}\lvert f(\theta)\rvert \mathrm{d}\theta \leq 4.$
This was solved independently by Kristiansen [Kr74] (only in the case when $c_k\in\mathbb{R}$) and Saff and Sheil-Small [SSS73] (for general $c_k\in \mathbb{C}$).
Additional thanks to: Winston Heap, Vjekoslav Kovac, Karlo Lelas
OPEN
Is there an entire non-linear function $f$ such that, for all $x\in\mathbb{R}$, $x$ is rational if and only if $f(x)$ is?
More generally, if $A,B\subseteq \mathbb{R}$ are two countable dense sets then is there an entire function such that $f(A)=B$?
Additional thanks to: Boris Alexeev and Dustin Mixon
SOLVED
Let $f=\sum_{n=0}^\infty a_nz^n$ be an entire function. Is it true that if $\lim_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}$ exists then it must be $0$?
Clunie (unpublished) proved this if $a_n\geq 0$ for all $n$. This was disproved in general by Clunie and Hayman [ClHa64], who showed that the limit can take any value in $[0,1/2]$.

SOLVED
Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $\pm 1$, such that $\sqrt{n} \ll \lvert P(z) \rvert \ll \sqrt{n}$ for all $\lvert z\rvert =1$, with the implied constants independent of $z$ and $n$?
Originally a conjecture of Littlewood. The answer is yes (for all $n\geq 2$), proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST19].

OPEN
Let $(S_n)_{n\geq 1}$ be a sequence of sets of complex numbers, none of which has a finite limit point. Does there exist an entire function $f(z)$ such that, for all $n\geq 1$, there exists some $k_n\geq 0$ such that $f^{(k_n)}(z) = 0\textrm{ for all }z\in S_n?$
SOLVED
Let $P(z)=\sum_{1\leq k\leq n}a_kz^k$ for some $a_k\in \mathbb{C}$ with $\lvert a_k\rvert=1$ for $1\leq k\leq n$. Does there exist a constant $c>0$ such that, for $n\geq 2$, we have $\max_{\lvert z\rvert=1}\lvert P(z)\rvert \geq (1+c)\sqrt{n}?$
The lower bound of $\sqrt{n}$ is trivial from Parseval's theorem. The answer is no (contrary to Erdős' initial guess). Kahane [Ka80] constructed 'ultraflat' polynomials $P(z)=\sum a_kz^k$ with $\lvert a_k\rvert=1$ such that $P(z)=(1+o(1))\sqrt{n}$ uniformly for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$, where the $o(1)$ term $\to 0$ as $n\to \infty$.

For more details see the paper [BoBo09] of Bombieri and Bourgain and where Kahane's construction is improved to yield such a polynomial with $P(z)=\sqrt{n}+O(n^{\frac{7}{18}}(\log n)^{O(1)})$ for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$.

SOLVED
Let $S$ be a string of length $2^k-1$ formed from an alphabet of $k$ characters. Must $S$ contain an abelian square: two consecutive blocks $x$ and $y$ such that $y$ is a permutation of $x$?
Erdős initially conjectured that the answer is yes for all $k\geq 2$, but for $k=4$ this was disproved by de Bruijn and Erdős. At least, this is what Erdős writes, but gives no construction or reference, and a simple computer search produces no such counterexamples for $k=4$. Perhaps Erdős meant $2^k$, where indeed there is an example for $k=4$: $1213121412132124.$

Erdős then asked if there is in fact an infinite string formed from $\{1,2,3,4\}$ which contains no abelian squares? This is equivalent to [192], and such a string was constructed by Keränen [Ke92]. The existence of this infinite string gives a negative answer to the problem for all $k\geq 4$.

Containing no abelian squares is a stronger property than being squarefree (the existence of infinitely long squarefree strings over alphabets with $k\geq 3$ characters was established by Thue).

We refer to a recent survey by Fici and Puzynina [FiPu23] for more background and related results.

SOLVED
For $A\subset \mathbb{R}^2$ we define the upper density as $\overline{\delta}(A)=\limsup_{R\to \infty}\frac{\lambda(A \cap B_R)}{\lambda(B_R)},$ where $\lambda$ is the Lebesgue measure and $B_R$ is the ball of radius $R$.

Estimate $m_1=\sup \overline{\delta}(A),$ where $A$ ranges over all measurable subsets of $\mathbb{R}^2$ without two points distance $1$ apart. In particular, is $m_1\leq 1/4$?

A question of Moser [Mo66]. A lower bound of $m_1\geq \pi/8\sqrt{3}\approx 0.2267$ is given by taking the union of open circular discus of radius $1/2$ at a regular hexagonal lattice suitably spaced aprt. Croft [Cr67] gives a small improvement of $m_1\geq 0.22936$.

The trivial upper bound is $m_1\leq 1/2$, since for any unit vector $u$ the sets $A$ and $A+u$ must be disjoint. Erdős' question was solved by Ambrus, Csiszárik, Matolcsi, Varga, and Zsámboki [ACMVZ23] who proved that $m_1\leq 0.247$.

OPEN
Let $d_n=p_{n+1}-p_n$. Prove that $\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.$
Cramer proved an upper bound of $O(N(\log N)^4)$ conditional on the Riemann hypothesis. The prime number theorem immediately implies a lower bound of $\gg N(\log N)^2$.
OPEN
For every $c\geq 0$ the density $f(c)$ of integers for which $\frac{p_{n+1}-p_n}{\log n}< c$ exists and is a continuous function of $c$.
OPEN
Let $N_k=2\cdot 3\cdots p_k$ and $\{a_1<a_2<\cdots <a_{\phi(N_k)}\}$ be the integers $<N_k$ which are relatively prime to $N_k$. Then, for any $c\geq 0$, the limit $\frac{\#\{ a_i-a_{i-1}\leq c \frac{N_k}{\phi(N_k)} : 2\leq i\leq \phi(N_k)\}}{\phi(N_k)}$ exists and is a continuous function of $c$.
OPEN
Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$. Is it true that $f(n)=o(\log n)$?
Erdős [Er50] proved that there are infinitely many $n$ such that $f(n)\gg \log\log n$. Erdős could not even prove that there do not exist infinitely many integers $n$ such that for all $2^k<n$ the number $n-2^k$ is prime (probably $n=105$ is the largest such integer).

SOLVED
Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap \{1,\ldots,N\}\rvert \gg \log N$ for all large $N$. Let $f(n)$ count the number of solutions to $n=p+a$ for $p$ prime and $a\in A$. Is it true that $\limsup f(n)=\infty$?
Erdős [Er50] proved this when $A=\{2^k : k\geq 0\}$. Solved by Chen and Ding [ChDi22].

OPEN
Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\log x$ many consecutive primes $\leq x$ such that the difference between any two is $>c_2$?
This is well-known if $c_1$ is sufficiently small.
SOLVED
Let $f:\mathbb{N}\to \{-1,1\}$ be a multiplicative function. Is it true that $\lim_{N\to \infty}\frac{1}{N}\sum_{n\leq N}f(n)$ always exists?
Wintner observed that if $f$ can take complex values on the unit circle then the limit need not exist. The answer is yes, as proved by Wirsing [Wi67], and generalised by Halász [Ha68].
OPEN
Is there an infinite set of primes $P$ such that if $\{a_1<a_2<\cdots\}$ is the set of integers divisible only by primes in $P$ then $\lim a_{i+1}-a_i=\infty$?
Originally asked to Erdős by Wintner. The limit is infinite for a finite set of primes, which follows from a theorem of Pólya.
Additional thanks to: Boris Alexeev and Dustin Mixon
OPEN - $100 Let$f(N)$be the maximum size of$A\subseteq \{1,\ldots,N\}$such that the sums$a+b+c$with$a,b,c\in A$are all distinct (aside from the trivial coincidences). Is it true that $f(N)\sim N^{1/3}?$ Originally asked to Erdős by Bose. Bose and Chowla [BoCh62] provided a construction proving one half of this, namely $(1+o(1))N^{1/3}\leq f(N).$ The best upper bound known to date is due to Green [Gr01], $f(N) \leq ((7/2)^{1/3}+o(1))N^{1/3}$ (note that$(7/2)^{1/3}\approx 1.519\cdots$). More generally, Bose and Chowla conjectured that the maximum size of$A\subseteq \{1,\ldots,N\}$with all$r$-fold sums distinct (aside from the trivial coincidences) then $\lvert A\rvert \sim N^{1/r}.$ This is known only for$r=2$(see [30]). Additional thanks to: Cedric Pilatte OPEN For every$n\geq 2$there exist distinct integers$1\leq x<y<z$such that $\frac{4}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ The Erdős-Straus conjecture. The existence of a representation of$4/n$as the sum of at most four distinct unit fractions follows trivially from a greedy algorithm. Schinzel conjectured the generalisation that, for any fixed$a$, if$n$is sufficiently large in terms of$a$then there exist distinct integers$1\leq x<y<z$such that $\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ OPEN Let$a_1<a_2<\cdots$be a sequence of integers such that $\lim_{n\to \infty}\frac{a_n}{a_{n-1}^2}=1$ and$\sum\frac{1}{a_n}\in \mathbb{Q}$. Then, for all sufficiently large$n\geq 1$, $a_n = a_{n-1}^2-a_{n-1}+1.$ A sequence defined in such a fashion is known as Sylvester's sequence. OPEN Let$C>1$. Does the set of integers of the form$p+\lfloor C^k\rfloor$, for some prime$p$and$k\geq 0$, have density$>0$? Originally asked to Erdős by Kalmár. Erdős believed the answer is yes. Romanoff [Ro34] proved that the answer is yes if$C$is an integer. SOLVED Let$A\subseteq \mathbb{N}$be an infinite set such that$\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that $\limsup_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}\geq 3?$ Erdős writes it is 'easy to see' that this holds with$3$replaced by$2$, and that$3$would be best possible here. We do not see an easy argument that this holds with$2$, but this follows e.g. from the main result of Mann [Ma60]. The answer is yes, proved by Freiman [Fr73]. Additional thanks to: Zachary Chase SOLVED Let$(a,b)=1$. Every large integer is the sum of distinct integers of the form$a^kb^l$with$k,l\geq 0$. Proved by Birch [Bi59]. OPEN Let$n_1<n_2<\cdots$be a sequence of integers such that $\limsup \frac{n_k}{k}=\infty.$ Is $\sum_{k=1}^\infty \frac{1}{2^{n_k}}$ transcendental? Erdős [Er75c] proved the answer is yes under the stronger condition that$\limsup n_k/k^t=\infty$for all$t\geq 1$. OPEN Are there infinitely many$n$such that, for all$k\geq 1$, $\omega(n+k) \ll k,$ where the implied constant depends only on$n$? (Here$\omega(n)$is the number of distinct prime divisors of$n$.) Related to [69]. Erdős and Graham [ErGr80] write 'we just know too little about sieves to be able to handle such a question ("we" here means not just us but the collective wisdom (?) of our poor struggling human race).' See also [679]. OPEN Is $\sum_n \frac{\phi(n)}{2^n}$ irrational? Here$\phi$is the Euler totient function. SOLVED Is $\sum \frac{\sigma(n)}{2^n}$ irrational? (Here$\sigma(n)$is the sum of divisors function.) The answer is yes, as shown by Nesterenko [Ne96]. OPEN Is $\sum \frac{p_n}{2^n}$ irrational? (Here$p_n$is the$n$th prime.) OPEN Let$k\geq 1$and$\sigma_k(n)=\sum_{d\mid n}d^k$. Is $\sum \frac{\sigma_k(n)}{n!}$ irrational? This is known now for$1\leq k\leq 4$. The cases$k=1,2$are reasonably straightforward, as observed by Erdős [Er52]. The case$k=3$was proved independently by Schlage-Puchta [ScPu06] and Friedlander, Luca, and Stoiciu [FLC07]. The case$k=4$was proved by Pratt [Pr22]. SOLVED Let$a_1<a_2<\cdots $be an infinite sequence of integers such that$a_{i+1}/a_i\to 1$. If every arithmetic progression contains infinitely many integers which are the sum of distinct$a_i$then every sufficiently large integer is the sum of distinct$a_i$. This was disproved by Cassels [Ca60]. OPEN Let$A\subseteq \mathbb{N}$be such that $\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty$ and $\sum_{n\in A} \{ \theta n\}=\infty$ for every$\theta\in (0,1)$, where$\{x\}$is the distance of$x$from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of$A$. Cassels [Ca60] proved this under the alternative hypotheses $\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert\gg \log\log x$ and $\sum_{n\in A} \{ \theta n\}^2=\infty$ for every$\theta\in (0,1)$. SOLVED Let$z_1,z_2,\ldots \in [0,1]$be an infinite sequence, and define the discrepancy $D_N(I) = \#\{ n\leq N : z_n\in I\} - N\lvert I\rvert.$ Must there exist some interval$I\subseteq [0,1]$such that $\limsup_{N\to \infty}\lvert D_N(I)\rvert =\infty?$ The answer is yes, as proved by Schmidt [Sc68], who later showed [Sc72] that in fact this is true for all but countably many intervals of the shape$[0,x]$. Essentially the best possible result was proved by Tijdeman and Wagner [TiWa80], who proved that, for almost all intervals of the shape$[0,x)$, we have $\limsup_{N\to \infty}\frac{\lvert D_N([0,x))\rvert}{\log N}\gg 1.$ Additional thanks to: Cedric Pilatte and Stefan Steinerberger OPEN Let$n\geq 1$and$f(n)$be maximal such that, for every$a_1\leq \cdots \leq a_n\in \mathbb{N}$we have $\max_{\lvert z\rvert=1}\left\lvert \prod_{i}(1-z^{a_i})\right\rvert\geq f(n).$ Estimate$f(n)$- in particular, is it true that there exists some constant$c>0$such that $f(n) \geq \exp(n^{c})?$ Erdős and Szekeres [ErSz59] proved that$\lim f(n)^{1/n}=1$and$f(n)>\sqrt{2n}$. Erdős proved an upper bound of$f(n) < \exp(n^{1-c})$for some constant$c>0$with probabilistic methods. Atkinson [At61] showed that$f(n) <\exp(cn^{1/2}\log n)$for some constant$c>0$. This was improved to $f(n) \leq \exp( cn^{1/3}(\log n)^{4/3})$ by Odlyzko [Od82]. If we denote by$f^*(n)$the analogous quantity with the assumption that$a_1<\cdots<a_n$then Bourgain and Chang [BoCh18] prove that $f^*(n)< \exp(c(n\log n)^{1/2}\log\log n).$ Additional thanks to: Zachary Chase, Stefan Steinerberger OPEN Let$A\subseteq \mathbb{N}$be an infinite set. Is $\sum_{n\in A}\frac{1}{2^n-1}$ irrational? If$A=\mathbb{N}$then this series is$\sum_{n}\frac{d(n)}{2^n}$, where$d(n)$is the number of divisors of$n$, which Erdős [Er48] proved is irrational. OPEN Let$a_n\to \infty$. Is $\sum_{n} \frac{d(n)}{a_1\cdots a_n}$ irrational, where$d(n)$is the number of divisors of$n$? Erdős and Straus [ErSt71] have proved this is true if$a_n$is monotone, i.e.$a_{n-1}\leq a_n$for all$n$. Erdős [Er48] proved that$\sum_n \frac{d(n)}{t^n}$is for any integer$t\geq 2$. OPEN Is the sum $\sum_{n} \mu(n)^2\frac{n}{2^n}$ irrational? Additional thanks to: Boris Alexeev and Dustin Mixon OPEN Let$a_n$be a sequence such that$a_n/n\to \infty$. Is the sum $\sum_n \frac{a_n}{2^{a_n}}$ irrational? This is true under either of the stronger assumptions that •$a_{n+1}-a_n\to \infty$or •$a_n \gg n\sqrt{\log n\log\log n}$. Erdős and Graham speculate that the condition$\limsup a_{n+1}-a_n=\infty$is not sufficient, but know of no example. OPEN Are there infinitely many$n$such that there exists some$t\geq 2$and$a_1,\ldots,a_t\geq 1$such that $\frac{n}{2^n}=\sum_{1\leq k\leq t}\frac{a_k}{2^{a_k}}?$ Is this true for all$n$? Is there a rational$x$such that $x = \sum_{k=1}^\infty \frac{a_k}{2^{a_k}}$ has at least two solutions? Related to [260]. Additional thanks to: Zachary Chase OPEN Suppose$a_1<a_2<\cdots$is a sequence of integers such that for all integer sequences$t_n$with$t_n\geq 1$the sum $\sum_{n=1}^\infty \frac{1}{t_na_n}$ is irrational. How slowly can$a_n$grow? One possible definition of an 'irrationality sequence' (see also [263] and [264]). An example of such a sequence is$a_n=2^{2^n}$, while a non-example is$a_n=n!$. It is known that if$a_n$is such a sequence then$a_n^{1/n}\to\infty$. OPEN Let$a_n$be a sequence of integers such that for every sequence of integers$b_n$with$b_n/a_n\to 1$the sum $\sum\frac{1}{b_n}$ is irrational. Is$a_n=2^{2^n}$such a sequence? Must such a sequence satisfy$a_n^{1/n}\to \infty$? One possible definition of an 'irrationality sequence' (see also [262] and [264]). OPEN Let$a_n$be a sequence of integers such that, for every bounded sequence$b_n$, the sum $\sum \frac{1}{a_n+b_n}$ is irrational. Are$a_n=2^n$or$a_n=n!$examples of such a sequence? Is there such a sequence with$a_n<n^k$? A possible definition of an 'irrationality sequence' (see also [262] and [263]). One example is$a_n=2^{2^n}$. OPEN How fast can$a_n\to \infty$grow if $\sum\frac{1}{a_n}\quad\textrm{and}\quad\sum\frac{1}{a_n-1}$ are both rational? Cantor observed that$a_n=\binom{n}{2}$is such a sequence. If we replace$-1$by a different constant then higher degree polynomials can be used - for example if we consider$\sum_{n\geq 2}\frac{1}{a_n}$and$\sum_{n\geq 2}\frac{1}{a_n-12}$then$a_n=n^3+6n^2+5n$is an example of both series being rational. Kovač [Ko24c] constructs a sequence$a_n$with this property which grows exponentially with$n$: $a_n > 1.01^n.$ Additional thanks to: Vjekoslav Kovac OPEN Let$a_n$be an infinite sequence of integers. There exists some integer$t\geq 1$such that $\sum \frac{1}{a_n+t}$ is irrational. This conjecture is due to Stolarsky. OPEN Let$F_1=F_2=1$and$F_{n+1}=F_n+F_{n-1}$be the Fibonacci sequence. Let$n_1<n_2<\cdots $be an infinite sequence with$n_{k+1}/n_k \geq c>1$. Must $\sum_k\frac{1}{F_{n_k}}$ be irrational? It may be sufficient to have$n_k/k\to \infty$. Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that$\sum \frac{1}{F_{2^n}}$is irrational. The nature of$\sum \frac{1}{F_n}$itself is unknown. SOLVED Let$X\subseteq \mathbb{R}^3$be the set of all points of the shape $\left( \sum_{n\in A} \frac{1}{n},\sum_{n\in A}\frac{1}{n+1},\sum_{n\in A} \frac{1}{n+2}\right)$ as$A\subseteq\mathbb{N}$ranges over all infinite sets with$\sum_{n\in A}\frac{1}{n}<\infty$. Does$X$contain an open set? Erdős and Straus proved the answer is yes for the 2-dimensional version, where$X\subseteq \mathbb{R}^2$is the set of $\left( \sum_{n\in A} \frac{1}{n},\sum_{n\in A}\frac{1}{n+1}\right)$ as$A\subseteq\mathbb{N}$ranges over all infinite sets with$\sum_{n\in A}\frac{1}{n}<\infty$. The answer is yes, proved by Kovač [Ko24], who constructs an explicit open ball inside the set. The analogous question for higher dimensions remains open. OPEN Let$P$be a finite set of primes with$\lvert P\rvert \geq 2$and let$\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P\}$. Is the sum $\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]},$ where$[a_1,\ldots,a_n]$is the lowest common multiple of$a_1,\ldots,a_n$, rational or irrational? If$P$is infinite this sum is always irrational. OPEN Let$f(n)\to \infty$as$n\to \infty$. Is it true that $\sum_n \frac{1}{(n+1)\cdots (n+f(n))}$ is irrational? Erdős and Graham write 'the answer is almost surely in the affirmative if$f(n)$is assumed to be nondecreasing'. Even the case$f(n)=n$is unknown, although Hansen [Ha75] has shown that $\sum_n \frac{1}{\binom{2n}{n}}=\sum_n \frac{n!}{(n+1)\cdots (n+n)}=\frac{1}{3}+\frac{2\pi}{3^{5/2}}$ is transcendental. OPEN For any$n$, let$A(n)=\{0<n<\cdots\}$be the infinite sequence with$a_0=0$and$a_1=n$and$a_{k+1}$is the least integer such that there is no three-term arithmetic progression in$\{a_0,\ldots,a_{k+1}\}$. Can the$a_k$be explicitly determined? How fast do they grow? It is easy to see that$A(1)$is the set of integers which have no 2 in their base 3 expansion. Odlyzko and Stanley have found similar characterisations are known for$A(3^k)$and$A(2\cdot 3^k)$for any$k\geq 0$, see [OdSt78]. There are no conjectures for the general case. OPEN Let$N\geq 1$. What is the largest$t$such that there are$A_1,\ldots,A_t\subseteq \{1,\ldots,N\}$with$A_i\cap A_j$a non-empty arithmetic progression for all$i\neq j$? Simonovits and Sós [SiSo81] have shown that$t\ll N^2$. It is possible that the maximal$t$is achieved when we take the$A_i$to be all arithmetic progressions in$\{1,\ldots,N\}$containing some fixed element. If we drop the non-empty requirement then Simonovits, Sós, and Graham [SiSoGr80] have shown that $t\leq \binom{N}{3}+\binom{N}{2}+\binom{N}{1}+1$ and this is best possible. Additional thanks to: Zachary Hunter OPEN Is there a covering system all of whose moduli are of the form$p-1$for some primes$p\geq 5$? Selfridge has found an example using divisors of$360$if$p=3$is allowed. OPEN If$G$is an abelian group then can there exist an exact covering of$G$by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets) A problem of Herzog and Schönheim. In [Er97c] Erdős asks this for finite (not necessarily abelian) groups. Additional thanks to: Boris Alexeev and Dustin Mixon SOLVED If a finite system of$r$congruences$\{ a_i\pmod{n_i} : 1\leq i\leq r\}$covers$2^r$consecutive integers then it covers all integers. This is best possible as the system$2^{i-1}\pmod{2^i}$shows. This was proved indepedently by Selfridge and Crittenden and Vanden Eynden [CrVE70]. OPEN Is there an infinite Lucas sequence$a_0,a_1,\ldots,$where$(a_0,a_1)=1$and$a_{n+2}=a_{n+1}+a_n$for$n\geq 0$such that all$a_k$are composite, and yet no integer has a common factor with every term of the sequence? Whether such a composite Lucas sequence even exists was open for a while, but using covering systems Graham [Gr64] showed that $a_0 = 1786772701928802632268715130455793$ $a_1 = 1059683225053915111058165141686995$ generate such a sequence. This problem asks whether one can have a composite Lucas sequence without 'an underlying system of covering congruences responsible'. OPEN For which$n$is there a covering system$\{ a_d\pmod{d} : d\mid n\}$such that if$x\equiv a_{d_1}\pmod{d_1}$and$x\equiv a_{d_2}\pmod{d_2}$then$(d_1,d_2)=1$? The density of such$n$is zero. OPEN Let$A=\{n_1<\cdots<n_r\}$be a finite set of integers. What is the minimum value of the density of integers not hit by a suitable choice of congruences$a_i\pmod{n_i}$? Is the minimum achieved when all the$a_i$are equal? OPEN Let$k\geq 3$. Is there a choice of congruence classes$a_p\pmod{p}$for every prime$p$such that all except finitely many integers can be written as$a_p+tp$for some prime$p$and integer$t\geq k$? Even the case$k=3$seems difficult. This may be true with the primes replaced by any set$A\subseteq \mathbb{N}$such that $\lvert A\cap [1,N]\rvert \gg N/\log N$ and $\sum_{\substack{n\in A\\ n\leq N}}\frac{1}{n} -\log\log N\to \infty$ as$N\to \infty$. For$k=1$or$k=2$any set$A$such that$\sum_{n\in A}\frac{1}{n}=\infty$has this property. OPEN Let$n_1<n_2<\cdots $be an infinite sequence of integers with associated$a_i\pmod{n_i}$, such that for some$\epsilon>0$we have$n_k>(1+\epsilon)k\log k$for all$k$. Then $\#\{ m<n_k : m\not\equiv a_i\pmod{n_i} \textrm{ for }1\leq i\leq k\}\neq o(k).$ Erdős and Graham [ErGr80] suggest this 'may have to await improvements in current sieve methods' (of which there have been several since 1980). OPEN Let$n_1<n_2<\cdots$be an infinite sequence with associated congruence classes$a_i\pmod{n_i}$such that the set of integers not satisfying any of the congruences$a_i\pmod{n_i}$has density$0$. Is it true that for every$\epsilon>0$there exists some$k$such that the density of integers not satisfying any of the congruences$a_i\pmod{n_i}$for$1\leq i\leq k$is less than$\epsilon$? The latter condition is clearly sufficient, the problem is if it's also necessary. The assumption implies$\sum \frac{1}{n_i}=\infty$. OPEN Let$A\subseteq \mathbb{N}$be an infinite set and consider the following greedy algorithm for a rational$x\in (0,1)$: choose the minimal$n\in A$such that$n\geq 1/x$and repeat with$x$replaced by$x-\frac{1}{n}$. If this terminates after finitely many steps then this produces a representation of$x$as the sum of distinct unit fractions with denominators from$A$. Does this process always terminate if$x$has odd denominator and$A$is the set of odd numbers? More generally, for which pairs$x$and$A$does this process terminate? In 1202 Fibonacci observed that this process terminates for any$x$when$A=\mathbb{N}$. The problem when$A$is the set of odd numbers is due to Stein. Graham [Gr64b] has shown that$\frac{m}{n}$is the sum of distinct unit fractions with denominators$\equiv a\pmod{d}$if and only if $\left(\frac{n}{(n,(a,d))},\frac{d}{(a,d)}\right)=1.$ Does the greedy algorithm always terminate in such cases? Graham [Gr64c] has also shown that$x$is the sum of distinct unit fractions with square denominators if and only if$x\in [0,\pi^2/6-1)\cup [1,\pi^2/6)$. Does the greedy algorithm for this always terminate? Erdős and Graham believe not - indeed, perhaps it fails to terminate almost always. Additional thanks to: Zachary Hunter OPEN Let$p:\mathbb{Z}\to \mathbb{Z}$be a polynomial whose leading coefficient is positive and such that there exists no$d\geq 2$with$d\mid p(n)$for all$n\geq 1$. Is it true that, for all sufficiently large$m$, there exist integers$1\leq n_1<\cdots <n_k$such that $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ and $m=p(n_1)+\cdots+p(n_k)?$ Graham [Gr63] has proved this when$p(x)=x$. Cassels [Ca60] has proved that these conditions on the polynomial imply every sufficiently large integer is the sum of$p(n_i)$with distinct$n_i$. Burr has proved this if$p(x)=x^k$with$k\geq 1$and if we allow$n_i=n_j$. SOLVED Let$f(k)$be the maximal value of$n_1$such that there exist$n_1<n_2<\cdots <n_k$with $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Is it true that $f(k)=(1+o(1))\frac{k}{e-1}?$ The upper bound$f(k) \leq (1+o(1))\frac{k}{e-1}$is trivial since for any$u\geq 1$we have $\sum_{u\leq n\leq eu}\frac{1}{n}=1+o(1),$ and hence if$f(k)=u$then we must have$k\geq (e-1-o(1))u$. Essentially solved by Croot [Cr01], who showed that for any$N>1$there exists some$k\geq 1$and $N<n_1<\cdots <n_k \leq (e+o(1))N$ with$1=\sum \frac{1}{n_i}$. Additional thanks to: Zachary Hunter SOLVED Let$f(k)$be the minimal value of$n_k$such that there exist$n_1<n_2<\cdots <n_k$with $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Is it true that $f(k)=(1+o(1))\frac{e}{e-1}k?$ It is trivial that$f(k)\geq (1+o(1))\frac{e}{e-1}k$, since for any$u\geq 1$$\sum_{e\leq n\leq eu}\frac{1}{n}= 1+o(1),$ and so if$eu\approx f(k)$then$k\leq \frac{e-1}{e}f(k)$. Proved by Martin [Ma00]. Additional thanks to: Zachary Hunter SOLVED Let$k\geq 2$. Is it true that there exists an interval$I$of width$(e-1+o(1))k$and integers$n_1<\cdots<n_k\in I$such that $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?$ The answer is yes, proved by Croot [Cr01]. OPEN Let$k\geq 2$. Is it true that, for any distinct integers$n_1<\cdots <n_k$such that $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ we must have$\max(n_{i+1}-n_i)\geq 3$? The example$1=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$shows that$3$would be best possible here. The lower bound of$\geq 2$is equivalent to saying that$1$is not the sum of reciprocals of consecutive integers, proved by Erdős [Er32]. This conjecture would follow for all but at most finitely many exceptions if it were known that, for all large$N$, there exists a prime$p\in [N,2N]$such that$\frac{p+1}{2}$is also prime. OPEN Is it true that there are only finitely many pairs of intervals$I_1,I_2$such that $\sum_{n_1\in I_1}\frac{1}{n_1}+\sum_{n_2\in I_2}\frac{1}{n_2}\in \mathbb{N}?$ This is still open even if$\lvert I_2\rvert=1$. It is perhaps true with two intervals replaced by any$k$intervals. OPEN Is it true that, for all sufficiently large$k$, there exist intervals$I_1,\ldots,I_k$with$\lvert I_i\rvert \geq 2$for$1\leq i\leq k$such that $1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}?$ OPEN Let$a\geq 1$. Must there exist some$b>a$such that $\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and }\sum_{a\leq n\leq b+1}\frac{1}{n}=\frac{r_2}{s_2},$ with$(r_i,s_i)=1$and$s_2<s_1$? If so, how does this$b(a)$grow with$a$? For example, $\sum_{3\leq n\leq 5}\frac{1}{n} = \frac{47}{60}\textrm{ and }\sum_{3\leq n\leq 6}\frac{1}{n}=\frac{19}{20}.$ OPEN Let$n\geq 1$and define$L_n$to be the least common multiple of$\{1,\ldots,n\}$and$a_n$by $\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}.$ Is it true that$(a_n,L_n)=1$and$(a_n,L_n)>1$both occur for infinitely many$n$? Steinerberger has observed that the answer to the second question is trivially yes: for example, any$n$which begins with a$2$in base$3$has$3\mid (a_n,L_n)$. More generally, if the leading digit of$n$in base$p$is$p-1$then$p\mid (a_n,L_n)$. There is in fact a necessary and sufficient condition: a prime$p\leq n$divides$(a_n,L_n)$if and only if$p$divides the numerator of$1+\cdots+\frac{1}{k}$, where$k$is the leading digit of$n$in base$p$. This can be seen by writing $a_n = \frac{L_n}{1}+\cdots+\frac{L_n}{n}$ and observing that the right-hand side is congruent to$1+\cdots+1/k$modulo$p$. (The previous claim about$p-1$follows immediately from Wolstenholme's theorem.) This leads to a heuristic prediction (see for example a preprint of Shiu [Sh16]) of$\asymp\frac{x}{\log x}$for the number of$n\in [1,x]$such that$(a_n,L_n)=1$. In particular, there should be infinitely many$n$, but the set of such$n$should have density zero. Unfortunately this heuristic is difficult to turn into a proof. Additional thanks to: Stefan Steinerberger OPEN Let$A$be the set of$n\in \mathbb{N}$such that there exist$1\leq m_1<\cdots <m_k=n$with$\sum\tfrac{1}{m_i}=1$. Explore$A$. In particular, • Does$A$have density$1$? • What are those$n\in A$not divisible by any$d\in A$with$1<d<n$? Straus observed that$A$is closed under multiplication. Furthermore, it is easy to see that$A$does not contain any prime power. Additional thanks to: Zachary Hunter OPEN Let$k\geq 1$and let$v(k)$be the minimal integer which does not appear as some$n_i$in a solution to $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ with$1\leq n_1<\cdots <n_k$. Estimate the growth of$v(k)$. Results of Bleicher and Erdős [BlEr75] imply$v(k) \gg k!$. It may be that$v(k)$grows doubly exponentially in$\sqrt{k}$or even$k$. An elementary inductive argument shows that$n_k\leq ku_k$where$u_1=1$and$u_{i+1}=u_i(u_i+1)$, and hence $v(k) \leq kc_0^{2^k},$ where $c_0=\lim_n u_n^{1/2^n}=1.26408\cdots$ is the 'Vardi constant'. Additional thanks to: Zachary Hunter SOLVED Let$N\geq 1$and let$t(N)$be the least integer$t$such that there is no solution to $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ with$t=n_1<\cdots <n_k\leq N$. Estimate$t(N)$. Erdős and Graham [ErGr80] could show $t(N)\ll\frac{N}{\log N},$ but had no idea of the true value of$t(N)$. Solved by Liu and Sawhney [LiSa24] (up to$(\log\log N)^{O(1)}$), who proved that $\frac{N}{(\log N)(\log\log N)^3(\log\log\log N)^{O(1)}}\ll t(N) \ll \frac{N}{\log N}.$ OPEN Let$N\geq 1$and let$k(N)$denote the smallest$k$such that there exist$N\leq n_1<\cdots <n_k$with $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Is it true that $\lim_{N\to \infty} k(N)-(e-1)N=\infty?$ Erdős and Straus [ErSt71b] have proved the existence of some constant$c>0$such that $-c < k(N)-(e-1)N \ll \frac{N}{\log N}.$ SOLVED Let$N\geq 1$and let$k(N)$be maximal such that there are$k$disjoint$A_1,\ldots,A_k\subseteq \{1,\ldots,N\}$with$\sum_{n\in A_i}\frac{1}{n}=1$for all$i$. Estimate$k(N)$. Is it true that$k(N)=o(\log N)$? More generally, how many disjoint$A_i$can be found in$\{1,\ldots,N\}$such that the sums$\sum_{n\in A_i}\frac{1}{n}$are all equal? Using sunflowers it can be shown that there are at least$N\exp(-O(\sqrt{\log N}))$such sets. Hunter and Sawhney have observed that Theorem 3 of Bloom [Bl23] (coupled with the trivial greedy approach) implies that$k(N)=(1-o(1))\log N$. Additional thanks to: Zachary Hunter and Mehtaab Sawhney SOLVED Let$N\geq 1$. How many$A\subseteq \{1,\ldots,N\}$are there such that$\sum_{n\in A}\frac{1}{n}=1$? It was not even known for a long time whether this is$2^{cN}$for some$c<1$or$2^{(1+o(1))N}$. In fact the former is true, and the correct value of$c$is now known. • Steinerberger [St24] proved the relevant count is at most$2^{0.93N}$; • Independently, Liu and Sawhney [LiSa24] gave both upper and lower bounds, proving that the count is $2^{(0.91\cdots+o(1))N},$ where$0.91\cdots$is an explicit number defined as the solution to a certain integral equation; • Again independently this same asymptotic was proved (with a different proof) by Conlon, Fox, He, Mubayi, Pham, Suk, and Verstraëte [CFHMPSV24], who prove more generally, for any$x\in \mathbb{Q}_{>0}$, a similar expression for the number of$A\subseteq \{1,\ldots,N\}$such that$\sum_{n\in A}\frac{1}{n}=x$; • The above papers all appeared within weeks of each other in 2024; in 2017 a similar question (with$\leq 1$rather than$=1$) was asked on MathOverflow by Mikhail Tikhomirov and proofs of the correct asymptotic were sketched by Lucia, RaphaelB4, and js21. Additional thanks to: Zachary Chase SOLVED Does every set$A\subseteq \mathbb{N}$of positive density contain some finite$S\subset A$such that$\sum_{n\in S}\frac{1}{n}=1$? The answer is yes, proved by Bloom [Bl21]. SOLVED Is there an infinite sequence$a_1<a_2<\cdots $such that$a_{i+1}-a_i=O(1)$and no finite sum of$\frac{1}{a_i}$is equal to$1$? There does not exist such a sequence, which follows from the positive solution to [298] by Bloom [Bl21]. SOLVED Let$A(N)$denote the maximal cardinality of$A\subseteq \{1,\ldots,N\}$such that$\sum_{n\in S}\frac{1}{n}\neq 1$for all$S\subseteq A$. Estimate$A(N)$. Erdős and Graham believe the answer is$A(N)=(1+o(1))N$. Croot [Cr03] disproved this, showing the existence of some constant$c<1$such that$A(N)<cN$for all large$N$. It is trivial that$A(N)\geq (1-\frac{1}{e}+o(1))N$. Liu and Sawhney [LiSa24] have proved that$A(N)=(1-1/e+o(1))N$. Additional thanks to: Zachary Hunter OPEN What is the size of the largest$A\subseteq \{1,\ldots,N\}$such that for any$a,b_1,\ldots,b_k\in A$with$k\geq 2$we have $\frac{1}{a}\neq \frac{1}{b_1}+\cdots+\frac{1}{b_k}?$ Is there a constant$c>1/2$such that$\lvert A\rvert >cN$is possible for all large$N$? The example$A=(N/2,N]\cap \mathbb{N}$shows that$\lvert A\rvert\geq N/2$. Additional thanks to: Zachary Hunter OPEN Is it true that, for any$\delta>1/2$, if$N$is sufficiently large and$A\subseteq \{1,\ldots,N\}$has$\lvert A\rvert \geq \delta N$then there exist$a,b,c\in A$such that $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}.$ The colouring version of this is [303], which was solved by Brown and Rödl [BrRo91]. The possible alternative question, that if$A\subseteq \mathbb{N}$is a set of positive lower density then must there exist$a,b,c\in A$such that $\frac{1}{a}=\frac{1}{b}+\frac{1}{c},$ has a negative answer, taking for example$A$to be the union of$[5^k,(1+\epsilon)5^k]$for large$k$and sufficiently small$\epsilon>0$. This was observed by Hunter and Sawhney. Additional thanks to: Zachary Hunter and Mehtaab Sawhney SOLVED Is it true that in any finite colouring of the integers there exists a monochromatic solution to $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$ with distinct$a,b,c$? The density version of this is [302]. This colouring version is true, as proved by Brown and Rödl [BrRo91]. OPEN For integers$1\leq a<b$let$N(a,b)$denote the minimal$k$such that there exist integers$1<n_1<\cdots<n_k$with $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Estimate$N(b)=\max_{1\leq a<b}N(a,b)$. Is it true that$N(b) \ll \log\log b$? Erdős [Er50c] proved that $\log\log b \ll N(b) \ll \frac{\log b}{\log\log b}.$ The upper bound was improved by Vose [Vo85] to $N(b) \ll \sqrt{\log b}.$ One can also investigate the average of$N(a,b)$for fixed$b$, and it is known that $\frac{1}{b}\sum_{1\leq a<b}N(a,b) \gg \log\log b.$ Related to [18]. Additional thanks to: Zachary Hunter SOLVED For integers$1\leq a<b$let$D(a,b)$be the minimal value of$n_k$such that there exist integers$1\leq n_1<\cdots <n_k$with $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Estimate$D(b)=\max_{1\leq a<b}D(a,b)$. Is it true that $D(b) \ll b(\log b)^{1+o(1)}?$ Bleicher and Erdős [BlEr76] have shown that $D(b)\ll b(\log b)^2.$ If$b=p$is a prime then $D(p) \gg p\log p.$ This was solved by Yokota [Yo88], who proved that $D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2.$ This was improved by Liu and Sawhney [LiSa24] to $D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}.$ OPEN Let$a/b\in \mathbb{Q}_{>0}$with$b$squarefree. Are there integers$1<n_1<\cdots<n_k$, each the product of two distinct primes, such that $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?$ For$n_i$the product of three distinct primes, this is true when$b=1$, as proved by Butler, Erdős and Graham [BEG15] (this paper is perhaps Erdős' last paper, appearing 19 years after his death). OPEN Are there two finite sets of primes$P,Q$such that $1=\left(\sum_{p\in P}\frac{1}{p}\right)\left(\sum_{q\in Q}\frac{1}{q}\right)?$ Asked by Barbeau [Ba76]. Can this be done if we drop the requirement that all$p\in P$are prime and just ask for them to be relatively coprime, and similarly for$Q$? OPEN Let$N\geq 1$. What is the smallest integer not representable as the sum of distinct unit fractions with denominators from$\{1,\ldots,N\}$? Is it true that the set of integers not representable as such has the shape$[m,\infty)$for some$m$? SOLVED Let$N\geq 1$. How many integers can be written as the sum of distinct unit fractions with denominators from$\{1,\ldots,N\}$? Are there$o(\log N)$such integers? The answer to the second question is no: there are at least$(1-o(1))\log N$many such integers, which follows from a more precise result of Croot [Cr99], who showed that every integer at most $\leq \sum_{n\leq N}\frac{1}{n}-(\tfrac{9}{2}+o(1))\frac{(\log\log N)^2}{\log N}$ can be so represented. Additional thanks to: Zachary Hunter and Mehtaab Sawhney SOLVED Let$\alpha >0$and$N\geq 1$. Is it true that for any$A\subseteq \{1,\ldots,N\}$with$\lvert A\rvert \geq \alpha N$there exists some$S\subseteq A$such that $\frac{a}{b}=\sum_{n\in S}\frac{1}{n}$ with$a\leq b =O_\alpha(1)$? Liu and Sawhney [LiSa24] observed that the main result of Bloom [Bl21] implies a positive solution to this conjecture. They prove a more precise version, that if$(\log N)^{-1/7+o(1)}\leq \alpha \leq 1/2$then there is some$S\subseteq A$such that $\frac{a}{b}=\sum_{n\in S}\frac{1}{n}$ with$a\leq b \leq \exp(O(1/\alpha))$. They also observe that the dependence$b\leq \exp(O(1/\alpha))$is sharp. OPEN What is the minimal value of$\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$as$A$ranges over all subsets of$\{1,\ldots,N\}$which contain no$S$such that$\sum_{n\in S}\frac{1}{n}=1$? Is it $e^{-(c+o(1))N}$ for some constant$c\in (0,1)$? It is trivially at least$1/[1,\ldots,N]$. OPEN Does there exist some$c>0$such that, for any$K>1$, whenever$A$is a sufficiently large finite multiset of integers with$\sum_{n\in A}\frac{1}{n}>K$there exists some$S\subseteq A$such that $1-e^{-cK} < \sum_{n\in S}\frac{1}{n}\leq 1?$ Erdős and Graham knew this with$e^{-cK}$replaced by$c/K^2$. Additional thanks to: Mehtaab Sawhney OPEN Are there infinitely many solutions to $\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},$ where$m\geq 2$is an integer and$p_1<\cdots<p_k$are distinct primes? For example, $\frac{1}{2}+\frac{1}{3}=1-\frac{1}{6}.$ SOLVED Let$n\geq 1$and let$m$be minimal such that$\sum_{n\leq k\leq m}\frac{1}{k}\geq 1$. We define $\epsilon(n) = \sum_{n\leq k\leq m}\frac{1}{k}-1.$ How small can$\epsilon(n)$be? Is it true that $\liminf n^2\epsilon(n)=0?$ This is true, and shown by Lim and Steinerberger [LiSt24] who proved that there exist infinitely many$n$such that $\epsilon(n)n^2\ll \left(\frac{\log\log n}{\log n}\right)^{1/2}.$ Erdős and Graham (and also Lim and Steinerberger) believe that the exponent of$2$is best possible here, in that$\liminf \epsilon(n) n^{2+\delta}=\infty$for all$\delta>0$. OPEN Let$u_1=2$and$u_{n+1}=u_n^2-u_n+1$. Let$a_1<a_2<\cdots $be any other sequence with$\sum \frac{1}{a_k}=1$. Is it true that $\liminf a_n^{1/2^n}<\lim u_n^{1/2^n}=c_0=1.264085\cdots?$$c_0$is called the Vardi constant and the sequence$u_n$is Sylvester's sequence. In [ErGr80] this problem is stated with the sequence$u_1=1$and$u_{n+1}=u_n(u_n+1)$, but Quanyu Tang has pointed out this is probably an error (since with that choice we do not have$\sum \frac{1}{u_n}=1$). This question with Sylvester's sequence is the most natural interpretation of what they meant to ask. Additional thanks to: Quanyu Tang SOLVED Is it true that if$A\subset \mathbb{N}\backslash\{1\}$is a finite set with$\sum_{n\in A}\frac{1}{n}<2$then there is a partition$A=A_1\sqcup A_2$such that $\sum_{n\in A_i}\frac{1}{n}<1$ for$i=1,2$? This is not true if$A$is a multiset, for example$2,3,3,5,5,5,5$. This is not true in general, as shown by Sándor [Sa97], who observed that the proper divisors of$120$form a counterexample. More generally, Sándor shows that for any$n\geq 2$there exists a finite set$A\subseteq \mathbb{N}\backslash\{1\}$with$\sum_{k\in A}\frac{1}{k}<n$and no partition into$n$parts each of which has$\sum_{k\in A_i}\frac{1}{k}<1$. The minimal counterexample is$\{2,3,4,5,6,7,10,11,13,14,15\}$, found by Tom Stobart. Additional thanks to: Tom Stobart OPEN Is there some constant$c>0$such that for every$n\geq 1$there exists some$\delta_k\in \{-1,0,1\}$for$1\leq k\leq n$with $0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?$ Is it true that for sufficiently large$n$, for any$\delta_k\in \{-1,0,1\}$, $\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}$ whenever the left-hand side is not zero? Inequality is obvious for the second claim, the problem is strict inequality. This fails for small$n$, for example $\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.$ Additional thanks to: Zachary Chase OPEN Let$A\subseteq \mathbb{N}$be an infinite arithmetic progression and$f:A\to \{-1,1\}$be a non-constant function. Must there exist a finite$S\subset A$such that $\sum_{n\in S}\frac{f(n)}{n}=0?$ What about if$A$is an arbitrary set of positive density? What if$A$is the set of squares excluding$1$? Erdős and Straus [ErSt75] proved this when$A=\mathbb{N}$. Sattler [Sat75] proved this when$A$is the set of odd numbers. For the squares$1$must be excluded or the result is trivially false, since $\sum_{k\geq 2}\frac{1}{k^2}<1.$ OPEN What is the size of the largest$A\subseteq \{1,\ldots,N\}$such that there is a function$\delta:A\to \{-1,1\}$such that $\sum_{n\in A}\frac{\delta_n}{n}=0$ and $\sum_{n\in A'}\frac{\delta_n}{n}\neq 0$ for all$A'\subsetneq A$? OPEN Let$S(N)$count the number of distinct sums of the form$\sum_{n\in A}\frac{1}{n}$for$A\subseteq \{1,\ldots,N\}$. Estimate$S(N)$. Bleicher and Erdős [BlEr75] proved the lower bound $\frac{N}{\log N}\prod_{i=3}^k\log_iN\leq \frac{\log S(N)}{\log 2},$ valid for$k\geq 4$and$\log_kN\geq k$, and also [BlEr76b] proved the upper bound $\log S(N)\leq \log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right),$ valid for$r\geq 1$and$\log_{2r}N\geq 1$. (In these bounds$\log_in$denotes the$i$-fold iterated logarithm.) See also [321]. Additional thanks to: Boris Alexeev and Dustin Mixon OPEN What is the size of the largest$A\subseteq \{1,\ldots,N\}$such that all sums$\sum_{n\in S}\frac{1}{n}$are distinct for$S\subseteq A$? Let$R(N)$be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that $\frac{N}{\log N}\prod_{i=3}^k\log_iN\leq R(N)\leq \frac{1}{\log 2}\log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right),$ valid for any$k\geq 4$with$\log_kN\geq k$and any$r\geq 1$with$\log_{2r}N\geq 1$. (In these bounds$\log_in$denotes the$i$-fold iterated logarithm.) See also [320]. Additional thanks to: Boris Alexeev, Zachary Hunter, and Dustin Mixon OPEN Let$k\geq 3$and$A\subset \mathbb{N}$be the set of$k$th powers. What is the order of growth of$1_A^{(k)}(n)$, i.e. the number of representations of$n$as the sum of$k$many$k$th powers? Does there exist some$c>0$and infinitely many$n$such that $1_A^{(k)}(n) >n^c?$ Connected to Waring's problem. The famous Hypothesis$K$of Hardy and Littlewood was that$1_A^{(k)}(n)\leq n^{o(1)}$, but this was disproved by Mahler [Ma36] for$k=3$, who constructed infinitely many$n$such that $1_A^{(3)}(n)\gg n^{1/12}$ (where$A$is the set of cubes). Erdős believed Hypothesis$K$fails for all$k\geq 4$, but this is unknown. Hardy and Littlewood made the weaker Hypothesis$K^*$that for all$N$and$\epsilon>0$$\sum_{n\leq N}1_A^{(k)}(n)^2 \ll_\epsilon N^{1+\epsilon}.$ Erdős and Graham remark: 'This is probably true but no doubt very deep. However, it would suffice for most applications.' Independently Erdős [Er36] and Chowla proved that for all$k\geq 3$and infinitely many$n$$1_A^{(k)}(n) \gg n^{c/\log\log n}$ for some constant$c>0$(depending on$k$). OPEN Let$1\leq m\leq k$and$f_{k,m}(x)$denote the number of integers$\leq x$which are the sum of$m$many nonnegative$k$th powers. Is it true that $f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}$ for all$\epsilon>0$? Is it true that if$m<k$then $f_{k,m}(x) \gg x^{m/k}$ for sufficiently large$x$? This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'. The case$k=2$was resolved by Landau, who showed $f_{2,2}(x) \sim \frac{cx}{\sqrt{\log x}}$ for some constant$c>0$. For$k>2$it is not known if$f_{k,k}(x)=o(x)$. OPEN Does there exist a polynomial$f(x)\in\mathbb{Z}[x]$such that all the sums$f(a)+f(b)$with$a<b$nonnegative integers are distinct? Erdős and Graham describe this problem as 'very annoying'. Probably$f(x)=x^5$should work. OPEN Let$k\geq 3$and$f_{k,3}(x)$denote the number of integers$\leq x$which are the sum of three nonnegative$k$th powers. Is it true that $f_{k,3}(x) \gg x^{3/k}$ or even$\gg_\epsilon x^{3/k-\epsilon}$? Mahler and Erdős [ErMa38] proved that$f_{k,2}(x) \gg x^{2/k}$. For$k=3$the best known is due to Wooley [Wo15], $f_{3,3}(x) \gg x^{0.917\cdots}.$ OPEN Let$A\subset \mathbb{N}$be an additive basis of order$2$. Must there exist$B=\{b_1<b_2<\cdots\}\subseteq A$which is also a basis such that $\lim_{k\to \infty}\frac{b_k}{k^2}$ does not exist? Erdős originally asked whether this was true with$A=B$, but this was disproved by Cassels [Ca57]. OPEN Suppose$A\subseteq \{1,\ldots,N\}$is such that if$a,b\in A$then$a+b\nmid ab$. Can$A$be 'substantially more' than the odd numbers? What if$a,b\in A$with$a\neq b$implies$a+b\nmid 2ab$? Must$\lvert A\rvert=o(N)$? The connection to unit fractions comes from the observation that$\frac{1}{a}+\frac{1}{b}$is a unit fraction if and only if$a+b\mid ab$. SOLVED Suppose$A\subseteq\mathbb{N}$and$C>0$is such that$1_A\ast 1_A(n)\leq C$for all$n\in\mathbb{N}$. Can$A$be partitioned into$t$many subsets$A_1,\ldots,A_t$(where$t=t(C)$depends only on$C$) such that$1_{A_i}\ast 1_{A_i}(n)<C$for all$1\leq i\leq t$and$n\in \mathbb{N}$? Asked by Erdős and Newman. Nešetřil and Rödl have shown the answer is no for all$C$(source is cited as 'personal communication' in [ErGr80]). Erdős had previously shown the answer is no for$C=3,4$and infinitely many other values of$C$. OPEN Suppose$A\subseteq \mathbb{N}$is a Sidon set. How large can $\limsup_{N\to \infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}$ be? Erdős proved that$1/2$is possible and Krückeberg [Kr61] proved$1/\sqrt{2}$is possible. Erdős and Turán [ErTu41] have proved this$\limsup$is always$\leq 1$. The fact that$1$is possible would follow if any finite Sidon set is a subset of a perfect difference set. OPEN Suppose$A\subset\mathbb{N}$is a minimal basis with positive density. Is it true that, for any$n\in A$, the (upper) density of integers which cannot be represented without using$n$is positive? Asked by Erdős and Nathanson. SOLVED Let$A,B\subseteq \mathbb{N}$such that for all large$N$$\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}$ and $\lvert B\cap \{1,\ldots,N\}\rvert \gg N^{1/2}.$ Is it true that there are infinitely many solutions to$a_1-a_2=b_1-b_2\neq 0$with$a_1,a_2\in A$and$b_1,b_2\in B$? Ruzsa has observed that there is a simple counterexample: take$A$to be the set of numbers whose binary representation has only non-zero digits in even places, and$B$similarly but with non-zero digits only in odd places. It is easy to see$A$and$B$both grow like$\gg N^{1/2}$and yet for any$n\geq 1$there is exactly one solution to$n=a+b$with$a\in A$and$b\in B$. Ruzsa suggests that a non-trivial variant of this problem arises if one imposes the stronger condition that $\lvert A\cap \{1,\ldots,N\}\rvert \sim c_AN^{1/2}$ for some constant$c_A>0$as$N\to \infty$, and similarly for$B$. Additional thanks to: Imre Ruzsa OPEN Let$A\subseteq \mathbb{N}$and$D(A)$be the set of those numbers which occur infinitely often as$a_1-a_2$with$a_1,a_2\in A$. What conditions on$A$are sufficient to ensure$D(A)$has bounded gaps? Prikry, Tijdeman, Stewart, and others (see the survey articles [St78] and [Ti79]) have shown that a sufficient condition is that$A$has positive density. One can also ask what conditions are sufficient for$D(A)$to have positive density, or for$\sum_{d\in D(A)}\frac{1}{d}=\infty$, or even just$D(A)\neq\emptyset$. OPEN Let$A\subseteq \mathbb{N}$be a set of density zero. Does there exist a basis$B$such that$A\subseteq B+B$and $\lvert B\cap \{1,\ldots,N\}\rvert =o(N^{1/2})$ for all large$N$? Erdős and Newman [ErNe77] have proved this is true when$A$is the set of squares. See also [806]. OPEN Find the best function$f(n)$such that every$n$can be written as$n=a+b$where both$a,b$are$f(n)$-smooth (that is, are not divisible by any prime$p>f(n)$.) Erdős originally asked if even$f(n)\leq n^{1/3}$is true. This is known, and the best bound is due to Balog [Ba89] who proved that $f(n) \ll_\epsilon n^{\frac{4}{9\sqrt{e}}+\epsilon}$ for all$\epsilon>0$. (Note$\frac{4}{9\sqrt{e}}=0.2695\ldots$.) It is likely that$f(n)\leq n^{o(1)}$, or even$f(n)\leq e^{O(\sqrt{\log n})}$. Additional thanks to: Zachary Hunter OPEN Let$d(A)$denote the density of$A\subseteq \mathbb{N}$. Characterise those$A,B\subseteq \mathbb{N}$with positive density such that $d(A+B)=d(A)+d(B).$ One way this can happen is if there exists$\theta>0$such that $A=\{ n>0 : \{ n\theta\} \in X_A\}\textrm{ and }B=\{ n>0 : \{n\theta\} \in X_B\}$ where$\{x\}$denotes the fractional part of$x$and$X_A,X_B\subseteq \mathbb{R}/\mathbb{Z}$are such that$\mu(X_A+X_B)=\mu(X_A)+\mu(X_B)$. Are all possible$A$and$B$generated in a similar way (using other groups)? OPEN For$r\geq 2$let$h(r)$be the maximal finite$k$such that there exists a basis$A\subseteq \mathbb{N}$of order$r$(so every large integer is the sum of at most$r$integers from$A$) and exact order$k$(i.e.$k$is minimal such that every large integer is the sum of exactly$k$integers from$A$). Find the value of $\lim_r \frac{h(r)}{r^2}.$ Erdős and Graham [ErGr80b] have shown that a basis$A$has an exact order if and only if$a_2-a_1,a_3-a_2,a_4-a_3,\ldots$are coprime. They also prove that $\frac{1}{4}\leq \lim_r \frac{h(r)}{r^2}\leq \frac{5}{4}.$ It is known that$h(2)=4$, but even$h(3)$is unknown (it is$\geq 7$). Additional thanks to: Zachary Hunter SOLVED Let$A\subseteq \mathbb{N}$be an additive basis (of any finite order) such that$\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that $\lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}=\infty?$ The answer is no, and a counterexample was provided by Turjányi [Tu84]. This was generalised (to the replacement of$A+A$by the$h$-fold sumset$hA$for any$h\geq 2$) by Ruzsa and Turjányi [RT85]. Ruzsa and Turjányi do prove (under the same hypotheses) that $\lim_{N\to \infty}\frac{\lvert (A+A+A)\cap \{1,\ldots,3N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}=\infty,$ and conjecture that the same should be true with$(A+A)\cap \{1,\ldots,2N\}$in the numerator. Additional thanks to: Zachary Chase, Zach Hunter OPEN The restricted order of a basis is the least integer$t$(if it exists) such that every large integer is the sum of at most$t$distinct summands from$A$. What are necessary and sufficient conditions that this exists? Can it be bounded (when it exists) in terms of the order of the basis? What are necessary and sufficient conditions that this is equal to the order of the basis? Bateman has observed that for$h\geq 3$there is a basis of order$h$with no restricted order, taking $A=\{1\}\cup \{x>0 : h\mid x\}.$ Kelly [Ke57] has shown that any basis of order$2$has restricted order at most$4$and conjectures it always has restricted order at most$3$. The set of squares has order$4$and restricted order$5$(see [Pa33]) and the set of triangular numbers has order$3$and restricted order$3$(see [Sc54]). Is it true that if$A\backslash F$is a basis for all finite sets$F$then$A$must have a restricted order? What if they are all bases of the same order? OPEN Let$A\subseteq \mathbb{N}$be a basis of order$r$. Must the set of integers representable as the sum of exactly$r$distinct elements from$A$have positive density? OPEN Let$A=\{1,2,4,8,13,21,31,45,66,81,97,\ldots\}$be the greedy Sidon sequence: we begin with$1$and iteratively include the next smallest integer that preserves the Sidon property. What is the order of growth of$A$? Is it true that $\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2-\epsilon}$ for all$\epsilon>0$and large$N$? Erdős and Graham [ErGr80] also ask about the difference set$A-A$, whether this has positive density, and whether this contains$22$. This sequence is at OEIS A005282. OPEN Let$A=\{a_1<\cdots<a_k\}$be a finite set of integers and extend it to an infinite sequence$\overline{A}=\{a_1<a_2<\cdots \}$by defining$a_{n+1}$for$n\geq k$to be the least integer exceeding$a_n$which is not of the form$a_i+a_j$with$i,j\leq n$. Is it true that the sequence of differences$a_{m+1}-a_m$is eventually periodic? An old problem of Dickson. Even a starting set as small as$\{1,4,9,16,25\}$requires thousands of terms before periodicity occurs. OPEN With$a_1=1$and$a_2=2$let$a_{n+1}$for$n\geq 2$be the least integer$>a_n$which can be expressed uniquely as$a_i+a_j$for$i<j\leq n$. What can be said about this sequence? Do infinitely many pairs$a,a+2$occur? Does this sequence eventually have periodic differences? Is the density$0$? A problem of Ulam. The sequence is $1,2,3,4,6,8,11,13,16,18,26,28,\ldots$ at OEIS A002858. SOLVED If$A\subseteq \mathbb{N}$is a multiset of integers such that $\lvert A\cap \{1,\ldots,N\}\rvert\gg N$ for all$N$then must$A$be subcomplete? That is, must $P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}$ contain an infinite arithmetic progression? A problem of Folkman. Folkman [Fo66] showed that this is true if $\lvert A\cap \{1,\ldots,N\}\rvert\gg N^{1+\epsilon}$ for some$\epsilon>0$and all$N$. The original question was answered by Szemerédi and Vu [SzVu06] (who proved that the answer is yes). This is best possible, since Folkman [Fo66] showed that for all$\epsilon>0$there exists a multiset$A$with $\lvert A\cap \{1,\ldots,N\}\rvert\ll N^{1+\epsilon}$ for all$N$, such that$A$is not subcomplete. Additional thanks to: Zachary Hunter SOLVED If$A\subseteq \mathbb{N}$is a set of integers such that $\lvert A\cap \{1,\ldots,N\}\rvert\gg N^{1/2}$ for all$N$then must$A$be subcomplete? That is, must $P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}$ contain an infinite arithmetic progression? Folkman proved this under the stronger assumption that $\lvert A\cap \{1,\ldots,N\}\rvert\gg N^{1/2+\epsilon}$ for some$\epsilon>0$. This is true, and was proved by Szemerédi and Vu [SzVu06]. The stronger conjecture that this is true under $\lvert A\cap \{1,\ldots,N\}\rvert\geq (2N)^{1/2}$ seems to be still open (this would be best possible as shown by [Er61b]. OPEN Let$A\subseteq \mathbb{N}$be a complete sequence, and define the threshold of completeness$T(A)$to be the least integer$m$such that all$n\geq m$are in $P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}$ (the existence of$T(A)$is guaranteed by completeness). Is it true that there are infinitely many$k$such that$T(n^k)>T(n^{k+1})$? Erdős and Graham [ErGr80] remark that very little is known about$T(A)$in general. It is known that $T(n)=1, T(n^2)=128, T(n^3)=12758,$ $T(n^4)=5134240,\textrm{ and }T(n^5)=67898771.$ Erdős and Graham remark that a good candidate for the$n$in the question are$k=2^t$for large$t$, perhaps even$t=3$, because of the highly restricted values of$n^{2^t}$modulo$2^{t+1}$. OPEN Let$A=\{a_1< a_2<\cdots\}$be a set of integers such that •$A\backslash B$is complete for any finite subset$B$and •$A\backslash B$is not complete for any infinite subset$B$. Is it true that if$a_{n+1}/a_n \geq 1+\epsilon$for some$\epsilon>0$and all$n$then $\lim_n \frac{a_{n+1}}{a_n}=\frac{1+\sqrt{5}}{2}?$ Graham [Gr64d] has shown that the sequence$a_n=F_n-(-1)^{n}$, where$F_n$is the$n$th Fibonacci number, has these properties. Erdős and Graham [ErGr80] remark that it is easy to see that if$a_{n+1}/a_n>\frac{1+\sqrt{5}}{2}$then the second property is automatically satisfied, and that it is not hard to construct very irregular sequences satisfying both properties. OPEN Is there a sequence$A=\{a_1\leq a_2\leq \cdots\}$of integers with $\lim \frac{a_{n+1}}{a_n}=2$ such that $P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}$ has density$1$for every cofinite subsequence$A'$of$A$? OPEN For what values of$0\leq m<n$is there a complete sequence$A=\{a_1\leq a_2\leq \cdots\}$of integers such that •$A$remains complete after removing any$m$elements, but •$A$is not complete after removing any$n$elements? The Fibonacci sequence$1,1,2,3,5,\ldots$shows that$m=1$and$n=2$is possible. The sequence of powers of$2$shows that$m=0$and$n=1$is possible. The case$m=2$and$n=3$is not known. OPEN For what values of$t,\alpha \in (0,\infty)$is the sequence$\lfloor t\alpha^n\rfloor$complete? Even in the range$t\in (0,1)$and$\alpha\in (1,2)$the behaviour is surprisingly complex. For example, Graham [Gr64e] has shown that for any$k$there exists some$t_k\in (0,1)$such that the set of$\alpha$such that the sequence is complete consists of at least$k$disjoint line segments. It seems likely that the sequence is complete for all$t>0$and all$1<\alpha < \frac{1+\sqrt{5}}{2}$. Proving this seems very difficult, since we do not even known whether$\lfloor (3/2)^n\rfloor$is odd or even infinitely often. SOLVED If$A\subset\mathbb{N}$is a finite set of integers all of whose subset sums are distinct then $\sum_{n\in A}\frac{1}{n}<2.$ This was proved by Ryavec. The stronger statement that, for all$s\geq 0$, $\sum_{n\in A}\frac{1}{n^s} <\frac{1}{1-2^{-s}},$ was proved by Hanson, Steele, and Stenger [HSS77]. See also [1]. OPEN Let$p(x)\in \mathbb{Q}[x]$. Is it true that $A=\{ p(n)+1/n : n\in \mathbb{N}\}$ is strongly complete, in the sense that, for any finite set$B$, $\left\{\sum_{n\in X}n : X\subseteq A\backslash B\textrm{ finite }\right\}$ contains all sufficiently large rational numbers? Graham [Gr64f] proved this is true when$p(n)=n$. Erdős and Graham also ask which rational functions$r(x)\in\mathbb{Z}[x]$force$\{ r(n) : n\in\mathbb{N}\}$to be complete? OPEN Is there some$c>0$such that every measurable$A\subseteq \mathbb{R}^2$of measure$\geq c$contains the vertices of a triangle of area 1? Erdős (unpublished) proved that this is true if$A$has infinite measure, or if$A$is an unbounded set of positive measure. Additional thanks to: Vjekoslav Kovac OPEN Let$A\subseteq \mathbb{R}^2$be a measurable set with infinite measure. Must$A$contain the vertices of an isosceles trapezoid of area$1$? Erdős and Mauldin (unpublished) claim that this is true for trapezoids in general, but fails for parallelograms. Additional thanks to: Vjekoslav Kovac OPEN Let$\alpha,\beta\in \mathbb{R}_{>0}$such that$\alpha/\beta$is irrational. Is $\{ \lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 4\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 4\beta\rfloor,\ldots\}$ complete? What if$2$is replaced by some$\gamma\in(1,2)$? SOLVED Let$A\subseteq \mathbb{N}$be a lacunary sequence (so that$A=\{a_1<a_2<\cdots\}$and there exists some$\lambda>1$such that$a_{n+1}/a_n\geq \lambda$for all$n\geq 1$). Must $\left\{ \sum_{a\in A'}\frac{1}{a} : A'\subseteq A\textrm{ finite}\right\}$ contain all rationals in some open interval? Bleicher and Erdős conjecture the answer is no. Steinerberger has pointed out that as written this problem is trivial: simply take some lacunary$A$whose prime factors are restricted (e.g.$A=\{1,2,4,8,\ldots,\}$) - clearly any finite sum of the shape$\sum_{a\in A'}\frac{1}{a}$can only form a rational with denominator divisible by one of these restricted set of primes. This is puzzling, since Erdős and Graham were very aware of this kind of obstruction, so it's a strange thing to miss. I assume that there was some unwritten extra assumption intended (e.g.$A$contains a multiple of every integer). Additional thanks to: Stefan Steinerberger SOLVED Is there some$c>0$such that, for all sufficiently large$n$, there exist integers$a_1<\cdots<a_k\leq n$such that there are at least$cn^2$distinct integers of the form$\sum_{u\leq i\leq v}a_i$? This fails for$a_i=i$for example. Erdős and Graham also ask what happens if we drop the monotonicity restriction and just ask that the$a_i$are distinct. Perhaps some permutation of$\{1,\ldots,n\}$has at least$cn^2$such distinct sums (this was solved by Konieczny [Ko15], see [34]). The original problem was solved (in the affirmative) by Beker [Be23b]. They also ask how many consecutive integers$>n$can be represented as such a sum? Is it true that, for any$c>0$at least$cn$such integers are possible (for sufficiently large$n$)? See also [34] and [357]. Additional thanks to: Adrian Beker OPEN Let$1\leq a_1<\cdots <a_k\leq n$be integers such that all sums of the shape$\sum_{u\leq i\leq v}a_i$are distinct. How large can$k$be? Must$k=o(n)$? Asked by Erdős and Harzheim. What if we remove the monotonicity and/or the distinctness constraint? Also what is the least$m$which is not a sum of the given form? Can it be much larger than$n$? Erdős and Harzheim can show that$\sum_{x<a_i<x^2}\frac{1}{a_i}\ll 1$. Is it true that$\sum_i \frac{1}{a_i}\ll 1$? See also [34] and [356]. OPEN Is there a sequence$A=\{1\leq a_1<a_2<\cdots\}$such that every integer is the sum of some finite number of consecutive elements of$A$? Can the number of representations of$n$in this form tend to infinity with$n$? Erdős and Moser [Mo63] considered the case when$A$is the set of primes, and conjectured that the$\limsup$of the number of such representations in this case is infinite. They could not even prove that the upper density of the set of integers representable in this form is positive. OPEN Let$a_1<a_2<\cdots$be an infinite sequence of integers such that$a_1=k$and$a_{i+1}$is the least integer which is not a sum of consecutive earlier$a_j$s. What can be said about the density of this sequence? A problem of MacMahon, studied by Andrews [An75]. SOLVED Let$f(n)$be minimal such that$\{1,\ldots,n\}$can be partitioned into$f(n)$classes so that$n$cannot be expressed as a sum of distinct elements from the same class. How fast does$f(n)$grow? Alon and Erdős [AlEr96] proved that$f(n) = n^{1/3+o(1)}$, and more precisely $\frac{n^{1/3}}{(\log n)^{4/3}}\ll f(n) \ll \frac{n^{1/3}}{(\log n)^{1/3}}(\log\log n)^{1/3}.$ Vu [Vu07] improved the lower bound to $f(n) \gg \frac{n^{1/3}}{\log n}.$ Conlon, Fox, and Pham [CFP21] determined the order of growth of$f(n)$up to a multiplicative constant, proving $f(n) \asymp \frac{n^{1/3}(n/\phi(n))}{(\log n)^{1/3}(\log\log n)^{2/3}}.$ Additional thanks to: Noga Alon OPEN Let$c>0$and$n$be some large integer. What is the size of the largest$A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$such that$n$is not a sum of a subset of$A$? Does this depend on$n$in an irregular way? OPEN Let$A\subseteq \mathbb{N}$be a finite set of size$N$. Is it true that, for any fixed$t$, there are $\ll \frac{2^N}{N^{3/2}}$ many$S\subseteq A$such that$\sum_{n\in S}n=t$? If we further ask that$\lvert S\rvert=l$(for any fixed$l$) then is the number of solutions $\ll \frac{2^N}{N^2},$ with the implied constant independent of$l$and$t$? Erdős and Moser proved the first bound with an additional factor of$(\log n)^{3/2}$. This was removed by Sárközy and Szemerédi [SaSz65], thereby answering the first question in the affirmative. Stanley [St80] has shown that this quantity is maximised when$A$is an arithmetic progression and$t=\tfrac{1}{2}\sum_{n\in A}n$. Additional thanks to: Zachary Chase OPEN When is the product of two or more disjoint blocks of consecutive integers a power? Is it true that there are only finitely many collections of disjoint intervals$I_1,\ldots,I_n$of size$\lvert I_i\rvert \geq 4$for$1\leq i\leq n$such that $\prod_{1\leq i\leq n}\prod_{n\in I_i}n$ is a square? Erdős and Selfridge have proved that the product of consecutive integers is never a power. The condition$\lvert I_i\rvert \geq 4$is necessary here, since Pomerance has observed that the product of $(2^{n-1}-1)2^{n-1}(2^{n-1}+1),$ $(2^n-1)2^n(2^n+1),$ $(2^{2n-1}-2)(2^{2n-1}-1)2^{2n-1},$ and $(2^{2n-2}-2)(2^{2n}-1)2^{2n}$ is a always a square. OPEN Are there any triples of consecutive positive integers all of which are powerful (i.e. if$p\mid n$then$p^2\mid n$)? Erdős originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation$x^2=8y^2+1$. Additional thanks to: Zachary Chase OPEN Do all pairs of consecutive powerful numbers come from solutions to Pell equations? Is the number of such pairs$\leq x$bounded by$(\log x)^{O(1)}$? Erdős originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation$x^2=8y^2+1$. OPEN Are there any 2-full$n$such that$n+1$is 3-full? That is, if$p\mid n$then$p^2\mid n$and if$p\mid n+1$then$p^3\mid n+1$. Erdős originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation$x^2=8y^2+1$. Note that$8$is 3-full and$9$is 2-full. Is the the only pair of such consecutive integers? OPEN Let$B_2(n)$be the 2-full part of$n$(that is,$B_2(n)=n/n'$where$n'$is the product of all primes that divide$n$exactly once). Is it true that, for every fixed$k\geq 1$, $\prod_{n\leq m<n+k}B_2(m) \ll n^{2+o(1)}?$ Or perhaps even$\ll_k n^2$? It would also be interesting to find upper and lower bounds for the analogous product with$B_r$for$r\geq 3$, where$B_r(n)$is the$r$-full part of$n$(that is, the product of prime powers$p^a \mid n$such that$p^{a+1}\nmid n$and$a\geq r$). Is it true that, for every fixed$r,k\geq 2$and$\epsilon>0$, $\limsup \frac{\prod_{n\leq m<n+k}B_r(m) }{n^{1+\epsilon}}\to\infty?$ OPEN How large is the largest prime factor of$n(n+1)$? Mahler [Ma35] showed that this is$\gg \log\log n$. Schinzel [Sc67b] observed that for infinitely many$n$it is$\leq n^{O(1/\log\log\log n)}$. The truth is probably$\gg (\log n)^2$for all$n$. OPEN Let$\epsilon>0$and$k\geq 2$. Is it true that, for all sufficiently large$n$, there is a sequence of$k$consecutive integers in$\{1,\ldots,n\}$all of which are$n^\epsilon$-smooth? Erdős and Graham state that this is open even for$k=2$and 'the answer should be affirmative but the problem seems very hard'. Unfortunately the problem is trivially true as written (simply taking$\{1,\ldots,k\}$and$n>k^{1/\epsilon}$). There are (at least) two possible variants which are non-trivial, and it is not clear which Erdős and Graham meant. Let$P$be the sequence of$k$consecutive integers sought for. The potential strengthenings which make this non-trivial are: • Each$m\in P$must be$m^\epsilon$-smooth. If this is the problem then the answer is yes, which follows from a result of Balog and Wooley [BaWo98]: for any$\epsilon>0$and$k\geq 2$there exist infinitely many$m$such that$m+1,\ldots,m+k$are all$m^\epsilon$-smooth. • Each$m\in P$must be in$[n/2,n]$(say). In this case a positive answer also follows from the result of Balog and Wooley [BaWo98] for infinitely many$n$, but the case of all sufficiently large$n$is open. See also [370]. Additional thanks to: Cedric Pilatte SOLVED Are there infinitely many$n$such that the largest prime factor of$n$is$<n^{1/2}$and the largest prime factor of$n+1$is$<(n+1)^{1/2}$? Pomerance has observed that if we replace$1/2$in the exponent by$1/\sqrt{e}-\epsilon$for any$\epsilon>0$then this is true for density reasons (since the density of integers$n$whose greatest prime factor is$\leq n^{1/\sqrt{e}}$is$1/2$). Steinerberger has pointed out this problem has a trivial solution: take$n=m^2-1$, and then it is obvious that the largest prime factor of$n$is$\leq m+1\ll n^{1/2}$and the largest prime factor of$n+1$is$\leq m\ll (n+1)^{1/2}$(these$\ll$can be replaced by$<$if we choose$m$such that$m,m+1$are both composite). Given that Erdős and Graham describe the above observation of Pomerance and explicitly say about this problem that 'we know very little about this', it is strange that such a trivial obstruction was overlooked. Perhaps the problem they intended was subtly different, and the problem in this form was the result of a typographical error, but I have no good guess what was intended here. See also [369]. Additional thanks to: Stefan Steinerberger OPEN Let$P(n)$denote the largest prime factor of$n$. Show that the set of$n$with$P(n+1)>P(n)$has density$1/2$. Conjectured by Erdős and Pomerance [ErPo78], who proved that this set and its complement both have positive upper density. In [Er79e] Erdős also asks whether, for every$\alpha>0$, the density of the set of$n$where $P(n+1)>P(n)n^\alpha$ exists. SOLVED Let$P(n)$denote the largest prime factor of$n$. There are infinitely many$n$such that$P(n)>P(n+1)>P(n+2)$. Conjectured by Erdős and Pomerance [ErPo78], who proved the analogous result for$P(n)<P(n+1)<P(n+2)$. Solved by Balog [Ba01], who proved that this is true for$\gg \sqrt{x}$many$n\leq x$(for all large$x$). OPEN Show that the equation $n! = a_1!a_2!\cdots a_k!,$ with$n-1>a_1\geq a_2\geq \cdots \geq a_k$, has only finitely many solutions. This would follow if$P(n(n+1))/\log n\to \infty$, where$P(m)$denotes the largest prime factor of$m$(see Problem [368]). Hickerson conjectured the largest solution is $16! = 14! 5!2!.$ The condition$a_1<n-1$is necessary to rule out the trivial solutions when$n=a_2!\cdots a_k!$. Surányi was the first to conjecture that the only non-trivial solution to$a!b!=n!$is$6!7!=10!$. OPEN For any$m\in \mathbb{N}$, let$F(m)$be the minimal$k\geq 2$(if it exists) such that there are$a_1<\cdots <a_k=m$with$a_1!\cdots a_k!$a square. Let$D_k=\{ m : F(m)=k\}$. What is the order of growth of$\lvert D_k\cap\{1,\ldots,n\}\rvert$for$3\leq k\leq 6$? For example, is it true that$\lvert D_6\cap \{1,\ldots,n\}\rvert \gg n$? Studied by Erdős and Graham [ErGr76] (see also [LSS14]). It is known, for example, that: • no$D_k$contains a prime, •$D_2=\{ n^2 : n>1\}$, •$\lvert D_3\cap \{1,\ldots,n\}\rvert = o(\lvert D_4\cap \{1,\ldots,n\}\rvert)$, • the least element of$D_6$is$527$, and •$D_k=\emptyset$for$k>6$. Additional thanks to: Zachary Chase OPEN Is it true that for any$n,k\geq 1$, if$n+1,\ldots,n+k$are all composite then there are distinct primes$p_1,\ldots,p_k$such that$p_i\mid n+i$for$1\leq i\leq k$? Note this is trivial when$k\leq 2$. Originally conjectured by Grimm. This is a very difficult problem, since it in particular implies$p_{n+1}-p_n <p_n^{1/2-c}$for some constant$c>0$, in particular resolving Legendre's conjecture. Grimm proved that this is true if$k\ll \log n/\log\log n$. Erdős and Selfridge improved this to$k\leq (1+o(1))\log n$. Ramachandra, Shorey, and Tijdeman [RST75] have improved this to $k\ll\left(\frac{\log n}{\log\log n}\right)^3.$ OPEN Are there infinitely many$n$such that$\binom{2n}{n}$is coprime to$105$? Erdős, Graham, Ruzsa, and Straus [EGRS75] have shown that, for any two primes$p$and$q$, there are infinitely many$n$such that$\binom{2n}{n}$is coprime to$pq$. OPEN Is there some absolute constant$C>0$such that $\sum_{p\leq n}1_{p\nmid \binom{2n}{2}}\frac{1}{p}\leq C$ for all$n$? A question of Erdős, Graham, Ruzsa, and Straus [EGRS75], who proved that if$f(n)$is the sum in question then $\lim_{x\to \infty}\frac{1}{x}\sum_{n\leq x}f(n) = \sum_{k=2}^\infty \frac{\log k}{2^k}=\gamma_0$ and $\lim_{x\to \infty}\frac{1}{x}\sum_{n\leq x}f(n)^2 = \gamma_0^2,$ so that for almost all integers$f(m)=\gamma_0+o(1)$. They also prove that, for all large$n$, $f(n) \leq c\log\log n$ for some constant$c<1$. (It is trivial from Mertens estimates that$f(n)\leq (1+o(1))\log\log n$.) A positive answer would imply that $\sum_{p\leq n}1_{p\mid \binom{2n}{n}}\frac{1}{p}=(1-o(1))\log\log n,$ and Erdős, Graham, Ruzsa, and Straus say there is 'no doubt' this latter claim is true. OPEN Let$r\geq 0$. Does the density of integers$n$for which$\binom{n}{k}$is squarefree for at least$r$values of$1\leq k<n$exist? Is this density$>0$? Erdős and Graham state they can prove that, for$k$fixed and large, the density of$n$such that$\binom{n}{k}$is squarefree is$o_k(1)$. They can also prove that there are infinitely many$n$such that$\binom{n}{k}$is not squarefree for$1\leq k<n$, and expect that the density of such$n$is positive. OPEN Let$S(n)$denote the largest integer such that, for all$1\leq k<n$, the binomial coefficient$\binom{n}{k}$is divisible by$p^{S(n)}$for some prime$p$(depending on$k$). Is it true that $\limsup S(n)=\infty?$ If$s(n)$denotes the largest integer such that$\binom{n}{k}$is divisible by$p^{s(n)}$for some prime$p$for at least one$1\leq k<n$then it is easy to see that$s(n)\to \infty$as$n\to \infty$(and in fact that$s(n) \asymp \log n$). OPEN We call an interval$[u,v]$'bad' if the greatest prime factor of$\prod_{u\leq m\leq v}m$occurs with an exponent greater than$1$. Let$B(x)$count the number of$n\leq x$which are contained in at least one bad interval. Is it true that $B(x)\sim \#\{ n\leq x: p\mid n\rightarrow p\leq n^{1/2}\}?$ Erdős and Graham only knew that$B(x) > x^{1-o(1)}$. Similarly, we call an interval$[u,v]$'very bad' if$\prod_{u\leq m\leq v}m$is powerful. The number of integers$n\leq x$contained in at least one very bad interval should be$\ll x^{1/2}$. In fact, it should be asymptotic to the number of powerful numbers$\leq x$. See also [382]. OPEN The product of more than two consecutive integers is never powerful (a number$n$is powerful if whenever$p\mid n$we have$p^2\mid n$). Conjectured by Erdős and Selfridge. There are infinitely many$n$such that$n(n+1)$is powerful (see [364]). OPEN Let$u\leq v$be such that the largest prime dividing$\prod_{u\leq m\leq v}m$appears with exponent at least$2$. Is it true that$v-u=v^{o(1)}$? Can$v-u$be arbitrarily large? Erdős and Graham report it follows from results of Ramachandra that$v-u\leq v^{1/2+o(1)}$. See also [380]. OPEN Is it true that for every$k$there are infinitely many primes$p$such that the largest prime divisor of $\prod_{0\leq i\leq k}(p^2+i)$ is$p$? SOLVED If$1<k<n-1$then$\binom{n}{k}$is divisible by a prime$p<n/2$(except$\binom{7}{3}=5\cdot 7$). A conjecture of Erdős and Selfridge. Proved by Ecklund [Ec69], who made the stronger conjecture that whenever$n>k^2$the binomial coefficient$\binom{n}{k}$is divisible by a prime$p<n/k$. They have proved the weaker inequality$p\ll n/k^c$for some constant$c>0$. Additional thanks to: Zachary Chase OPEN Let $F(n) = \max_{\substack{m<n\\ m\textrm{ composite}}} m+p(m),$ where$p(m)$is the least prime divisor of$m$. Is it true that$F(n)>n$for all sufficiently large$n$? Does$F(n)-n\to \infty$as$n\to\infty$? A question of Erdős, Eggleton, and Selfridge, who write that 'plausible conjectures on primes' imply that$F(n)\leq n$for only finitely many$n$, and in fact it is possible that this quantity is always at least$n+(1-o(1))\sqrt{n}$(note that it is trivially$\leq n+\sqrt{n}$). OPEN Can$\binom{n}{k}$be the product of consecutive primes infinitely often? For example $\binom{21}{2}=2\cdot 3\cdot 5\cdot 7.$ Erdős and Graham write that 'a proof that this cannot happen infinitely often for$\binom{n}{2}$seems hopeless; probably this can never happen for$\binom{n}{k}$if$3\leq k\leq n-3$.' OPEN Is there an absolute constant$c>0$such that, for all$1\leq k< n$, the binomial coefficient$\binom{n}{k}$has a divisor in$(cn,n]$? Erdős once conjectured that$\binom{n}{k}$must always have a divisor in$(n-k,n]$, but this was disproved by Schinzel and Erdős [Sc58]. Additional thanks to: Zachary Chase OPEN Can one classify all solutions of $\prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j)$ where$1<k_1<k_2$and$m_1+k_1\leq m_2$? Are there only finitely many solutions? More generally, if$k_1>2$then for fixed$a$and$b$$a\prod_{1\leq i\leq k_1}(m_1+i)=b\prod_{1\leq j\leq k_2}(m_2+j)$ should have only a finite number of solutions. What if one just requires that the products have the same prime factors, say when$k_1=k_2$? OPEN Is it true that for every$n$there is a$k$such that $\prod_{1\leq i\leq k}(n+i) \mid \prod_{1\leq i\leq k}(n+k+i)?$ Asked by Erdős and Straus. For example when$n=2$we have$k=4$: $3\cdot 4\cdot 5\cdot 6 \mid 7\cdot 8\cdot 9\cdot 10.$ No example was known for$n\geq 10$. OPEN Let$f(n)$be the minimal$m$such that $n! = a_1\cdots a_k$ with$n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant$c$such that $f(n)-2n \sim c\frac{n}{\log n}?$ Erdős, Guy, and Selfridge [EGS82] have shown that $f(n)-2n \asymp \frac{n}{\log n}.$ OPEN Let$t(n)$be the maximum$m$such that $n!=a_1\cdots a_n$ with$m=a_1\leq \cdots \leq a_n$. Obtain good upper bounds for$t(n)$. In particular does there exist some constant$c>0$such that $t(n) \leq \frac{n}{e}-c\frac{n}{\log n}$ for infinitely many$n$? Erdős, Selfridge, and Straus have shown that $\lim \frac{t(n)}{n}=\frac{1}{e}.$ Alladi and Grinstead [AlGr77] have obtained similar results when the$a_i$are restricted to prime powers. OPEN Let$A(n)$denote the least value of$t$such that $n!=a_1\cdots a_t$ with$a_1\leq \cdots \leq a_t\leq n^2$. Is it true that $A(n)=\frac{n}{2}-\frac{n}{2\log n}+o\left(\frac{n}{\log n}\right)?$ If we change the condition to$a_t\leq n$it can be shown that $A(n)=n-\frac{n}{\log n}+o\left(\frac{n}{\log n}\right)$ via a greedy decomposition (use$n$as often as possible, then$n-1$, and so on). Other questions can be asked for other restrictions on the sizes of the$a_t$. OPEN Let$f(n)$denote the minimal$m$such that $n! = a_1\cdots a_t$ with$a_1<\cdots <a_t=a_1+m$. What is the behaviour of$f(n)$? Erdős and Graham write that they do not even know whether$f(n)=1$infinitely often (i.e. whether a factorial is the product of two consecutive integers infinitely often). SOLVED Let$t_k(n)$denote the least$m$such that $n\mid (m+1)(m+2)\cdots (m+k).$ Is it true that $\sum_{1\leq n\leq N}t_2(n)=o(N)?$ The answer is yes, proved by Hall. It is probably true that the sum is$o(N/(\log N)^c)$for some constant$c>0$. Similar questions can be asked for other$k\geq 3$. Additional thanks to: Zachary Chase OPEN Prove that there are no$n,x,y$with$x+n\leq y$such that $\mathrm{lcm}(x+1,\ldots,x+n) = \mathrm{lcm}(y+1,\ldots,y+n).$ It follows from the Thue-Siegel theorem that, for$n$fixed, there are only finitely many potential such$x$and$y$. OPEN Is it true that for every$k$there exists$n$such that $\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?$ Erdős and Graham write that$n+1$always divides$\binom{2n}{n}$(indeed$\frac{1}{n+1}\binom{2n}{n}$is the$n$th Catalan number), but it is quite rare that$n$divides$\binom{2n}{n}$. Pomerance [Po14] has shown that for any$k\geq 0$there are infinitely many$n$such that$n-k\mid\binom{2n}{n}$, although the set of such$n$has upper density$<1/3$. Pomerance also shows that the set of$n$such that $\prod_{1\leq i\leq k}(n+i)\mid \binom{2n}{n}$ has density$1$. OPEN Are there only finitely many solutions to $\prod_i \binom{2m_i}{m_i}=\prod_j \binom{2n_j}{n_j}$ with the$m_i,n_j$distinct? OPEN Are the only solutions to $n!=x^2-1$ when$n=4,5,7$? The Brocard-Ramanujan conjecture. Erdős and Graham describe this as an old conjecture, and write it 'is almost certainly true but it is intractable at present'. Overholt [Ov93] has shown that this has only finitely many solutions assuming a weak form of the abc conjecture. OPEN Is it true that there are no solutions to $n! = x^k\pm y^k$ with$x,y,n\in \mathbb{N}$, with$xy>1$and$k>2$? Erdős and Obláth [ErOb37] proved this is true when$(x,y)=1$and$k\neq 4$. Pollack and Shapiro [PoSh73] proved this is true when$(x,y)=1$and$k=4$. The known methods break down without the condition$(x,y)=1$. Additional thanks to: Zachary Chase OPEN For any$k\geq 2$let$g_k(n)$denote the maximal value of $n-(a_1+\cdots+a_k)$ where$a_1,\ldots,a_k$are integers such that$a_1!\cdots a_k! \mid n!$. Can one show that $\sum_{n\leq x}g_k(n) \sim c_k x\log x$ for some constant$c_k$? Is it true that there is a constant$c_k$such that for almost all$n<x$we have $g_k(n)=c_k\log x+o(\log x)?$ Erdős and Graham write that it is easy to show that$g_k(n) \ll_k \log n$always, but the best possible constant is unknown. See also [401]. OPEN Is there some function$\omega(r)$such that$\omega(r)\to \infty$as$r\to\infty$, such that for all large$n$there exist$a_1,a_2$with $a_1+a_2> n+\omega(r)\log n$ such that$a_1!a_2! \mid n!2^n3^n\cdots p_r^n$? See also [400]. SOLVED Prove that, for any finite set$A\subset\mathbb{N}$, there exist$a,b\in A$such that $\mathrm{gcd}(a,b)\leq a/\lvert A\rvert.$ A conjecture of Graham [Gr70], who also conjectured that (assuming$A$itself has no common divisor) the only cases where equality is achieved are when$A=\{1,\ldots,n\}$or$\{L/1,\ldots,L/n\}$(where$L=\mathrm{lcm}(1,\ldots,n)$) or$\{2,3,4,6\}$. Proved for all sufficiently large sets (including the sharper version which characterises the case of equality) independently by Szegedy [Sz86] and Zaharescu [Za87]. Proved for all sets by Balasubramanian and Soundararajan [BaSo96]. SOLVED Does the equation $2^m=a_1!+\cdots+a_k!$ with$a_1<a_2<\cdots <a_k$have only finitely many solutions? Asked by Burr and Erdős. Frankl and Lin [Li76] independently showed that the answer is yes, and the largest solution is $2^7=2!+3!+5!.$ In fact Lin showed that the largest power of$2$which can divide a sum of distinct factorials containing$2$is$2^{254}$, and that there are only 5 solutions to$3^m=a_1!+\cdots+a_k!$(when$m=0,1,2,3,6$). See also [404]. OPEN Let$f(a,p)$be the largest$k$such that there are$a=a_1<\cdots<a_k$such that $p^k \mid (a_1!+\cdots+a_k!).$ Is$f(a,p)$bounded by some absolute constant? What if this constant is allowed to depend on$a$and$p$? Is there a prime$p$and an infinite sequence$a_1<a_2<\cdots$such that if$p^{m_k}$is the highest power of$p$dividing$\sum_{i\leq k}a_i!$then$m_k\to \infty$? See also [403]. Lin [Li76] has shown that$f(2,2) \leq 254$. OPEN Let$p$be a prime. Is it true that the equation $(p-1)!+a^{p-1}=p^k$ has only finitely many solutions? Erdős and Graham remark that it is probably true that in general$(p-1)!+a^{p-1}$is rarely a power at all (although this can happen, for example$6!+2^6=28^2$). OPEN Is it true that there are only finitely many powers of$2$which have only the digits$0$and$1$when written in base$3$? The only examples seem to be$4=1+3$and$256=1+3+3^2+3^5$. If we only allow the digits$1$and$2$then$2^{15}$seems to be the largest such power of$2$. This would imply via Kummer's theorem that $3\mid \binom{2^{k+1}}{2^k}$ for all large$k$. SOLVED Let$w(n)$count the number of solutions to $n=2^a+3^b+2^c3^d$ with$a,b,c,d\geq 0$integers. Is it true that$w(n)$is bounded by some absolute constant? A conjecture originally due to Newman. This is true, and was proved by Evertse, Györy, Stewart, and Tijdeman [EGST88]. OPEN Let$\phi(n)$be the Euler totient function and$\phi_k(n)$be the iterated$\phi$function, so that$\phi_1(n)=\phi(n)$and$\phi_k(n)=\phi(\phi_{k-1}(n))$. Let $f(n) = \min \{ k : \phi_k(n)=1\}.$ Does$f(n)/\log n$have a distribution function? Is$f(n)/\log n$almost always constant? What can be said about the largest prime factor of$\phi_k(n)$when, say,$k=\log\log n$? Pillai [Pi29] was the first to investigate this function, and proved $\log_3 n < f(n) < \log_2 n$ for all large$n$. Shapiro [Sh50] proved that$f(n)$is essentially multiplicative. Erdős, Granville, Pomerance, and Spiro [EGPS90] have proved that the answer to the first two questions is yes, conditional on a form of the Elliott-Halberstam conjecture. It is likely true that, if$k\to \infty$however slowly with$n$, then for almost$n$the largest prime factor of$\phi_k(n)$is$\leq n^{o(1)}$. Additional thanks to: Zachary Chase OPEN How many iterations of$n\mapsto \phi(n)+1$are needed before a prime is reached? Can infinitely many$n$reach the same prime? What is the density of$n$which reach any fixed prime? A problem of Finucane. One can also ask about$n\mapsto \sigma(n)-1$. OPEN Let$\sigma_1(n)=\sigma(n)$, the sum of divisors function, and$\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that $\lim_{k\to \infty} \sigma_k(n)^{1/k}=\infty?$ OPEN Let$g_1=g(n)=n+\phi(n)$and$g_k(n)=g(g_{k-1}(n))$. For which$n$and$r$is it true that$g_{k+r}(n)=2g_k(n)$for all large$k$? The known solutions to$g_{k+2}(n)=2g_k(n)$are$n=10$and$n=94$. Selfridge and Weintraub found solutions to$g_{k+9}(n)=9g_k(n)$and Weintraub found $g_{k+25}(3114)=729g_k(3114)$ for all$k\geq 6$. OPEN Let$\sigma_1(n)=\sigma(n)$, the sum of divisors function, and$\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that, for every$m,n$, there exist some$i,j$such that$\sigma_i(m)=\sigma_j(n)$? In [Er79d] Erdős attributes this conjecture to van Wijngaarden, who told it to Erdős in the 1950s. That is, there is (eventually) only one possible sequence that the iterated sum of divisors function can settle on. Selfridge reports numerical evidence which suggests the answer is no, but Erdős and Graham write 'it seems unlikely that anything can be proved about this in the near future'. See also [413] and [414]. OPEN Let$\omega(n)$count the number of distinct primes dividing$n$. Are there infinitely many$n$such that, for all$m<n$, we have$m+\omega(m) \leq n$? Can one show that there exists an$\epsilon>0$such that there are infinitely many$n$where$m+\epsilon \omega(m)\leq n$for all$m<n$? In [Er79] Erdős calls such an$n$a 'barrier' for$\omega$. Some other natural number theoretic functions (such as$\phi$and$\sigma$) have no barriers because they increase too rapidly. Erdős believed that$\omega$should have infinitely many barriers. In [Er79d] he proves that$F(n)=\prod k_i$, where$n=\prod p_i^{k_i}$, has infinitely many barriers (in fact the set of barriers has positive density in the integers). Erdős also believed that$\Omega$, the count of the number of prime factors with multiplicity), should have infinitely many barriers. Selfridge found the largest barrier for$\Omega$which is$<10^5$is$99840$. In [ErGr80] this problem is suggested as a way of showing that the iterated behaviour of$n\mapsto n+\omega(n)$eventually settles into a single sequence, regardless of the starting value of$n$(see also [412] and [414]). Erdős and Graham report it could be attacked by sieve methods, but 'at present these methods are not strong enough'. See also [647] and [679]. OPEN Let$h_1(n)=h(n)=n+\tau(n)$(where$\tau(n)$counts the number of divisors of$n$) and$h_k(n)=h(h_{k-1}(n))$. Is it true, for any$m,n$, there exist$i$and$j$such that$h_i(m)=h_j(n)$? Asked by Spiro. That is, there is (eventually) only one possible sequence that the iterations of$n\mapsto h(n)$can settle on. Erdős and Graham believed the answer is yes. Similar questions can be asked by the iterates of many other functions. See also [412] and [413]. OPEN For any$n$let$F(n)$be the largest$k$such that any of the$k!$possible ordering patterns appears in some sequence of$\phi(m+1),\ldots,\phi(m+k)$with$m+k\leq n$. Is it true that $F(n)=(c+o(1))\log\log\log n$ for some constant$c$? Is the first pattern which fails to appear always $\phi(m+1)>\phi(m+2)>\cdots \phi(m+k)?$ Is it true that 'natural' ordering which mimics what happens to$\phi(1),\ldots,\phi(k)$is the most likely to appear? Erdős [Er36b] proved that $F(n)\asymp \log\log\log n,$ and similarly if we replace$\phi$with$\sigma$or$\tau$or$\nu$or any 'decent' additive or multiplicative function. OPEN Let$V(x)$count the number of$n\leq x$such that$\phi(m)=n$is solvable. Does$V(2x)/V(x)\to 2$? Is there an asymptotic formula for$V(x)$? Pillai [Pi29] proved$V(x)=o(x)$. Erdős [Er35b] proved$V(x)=x(\log x)^{-1+o(1)}$. The behaviour of$V(x)$is now almost completely understood. Maier and Pomerance [MaPo88] proved $V(x)=\frac{x}{\log x}e^{(C+o(1))(\log\log\log x)^2},$ for some explicit constant$C>0$. Ford [Fo98] improved this to $V(x)\asymp\frac{x}{\log x}e^{C_1(\log\log\log x-\log\log\log\log x)^2+C_2\log\log\log x-C_3\log\log\log\log x}$ for some explicit constants$C_1,C_2,C_3>0$. Unfortunately this falls just short of an asymptotic formula for$V(x)$and determining whether$V(2x)/V(x)\to 2$. In [Er79e] Erdős asks further to estimate the number of$n\leq x$such that the smallest solution to$\phi(m)=n$satisfies$kx<m\leq (k+1)x$. See also [417]. Additional thanks to: Kevin Ford OPEN Let $V'(x)=\#\{\phi(m) : 1\leq m\leq x\}$ and $V(x)=\#\{\phi(m) \leq x : 1\leq m\}.$ Does$\lim V(x)/V'(x)$exist? Is it$>1$? It is trivial that$V'(x) \leq V(x)$. See also [416]. SOLVED Are there infinitely many integers not of the form$n-\phi(n)$? Asked by Erdős and Sierpiński. It follows from the Goldbach conjecture that every odd number can be written as$n-\phi(n)$. What happens for even numbers? Erdős [Er73b] has shown that a positive density set of integers cannot be written as$\sigma(n)-n$. This is true, as shown by Browkin and Schinzel [BrSc95], who show that any integer of the shape$2^{k}\cdot 509203$is not of this form. It seems to be open whether there is a positive density set of integers not of this form. Additional thanks to: Stefan Steinerberger SOLVED If$\tau(n)$counts the number of divisors of$n$, then what is the set of limit points of $\frac{\tau((n+1)!)}{\tau(n!)}?$ Erdős and Graham noted that any number of the shape$1+1/k$for$k\geq 1$is a limit point (and thus so too is$1$), but knew of no others. Mehtaab Sawhney has shared the following simple argument that proves that the above limit points are in fact the only ones. If$v_p(m)$is the largest$k$such that$p^k\mid m$then$\tau(m)=\prod_p (v_p(m)+1)$and so $\frac{\tau((n+1)!)}{\tau(n!)} = \prod_{p|n+1}\left(1+\frac{v_p(n+1)}{v_p(n!)+1}\right).$ Note that$v_p(n!)\geq n/p$, and furthermore$n+1$has$<\log n$prime divisors, each of which satisfy$v_p(n+1)<\log n$. It follows that the contribution from$p\leq n^{2/3}$is at most $\left(1+\frac{\log n}{n^{1/3}}\right)^{\log n}\leq 1+o(1).$ There is at most one$p\mid n+1$with$p\geq n^{2/3}$which (if present) contributes exactly $\left(1+\frac{1}{\frac{n+1}{p}}\right).$ We have proved the claim, since these two facts combined show that the ratio in question is either$1+o(1)$or$1+1/k+o(1)$, the latter occurring if$n+1=pk$for some$p>n^{2/3}$. Additional thanks to: Zachary Chase, Mehtaab Sawhney OPEN If$\tau(n)$counts the number of divisors of$n$then let $F(f,n)=\frac{\tau((n+\lfloor f(n)\rfloor)!)}{\tau(n!)}.$ Is it true that $\lim_{n\to \infty}F((\log n)^C,n)=\infty$ for large$C$? Is it true that$F(\log n,n)$is everywhere dense in$(1,\infty)$? More generally, if$f(n)\leq \log n$is a monotonic function then is$F(f,n)$everywhere dense? Erdős and Graham write that it is easy to show that$\lim F(n^{1/2},n)=\infty$, and in fact the$n^{1/2}$can be replaced by$n^{1/2-c}$for some small constant$c>0$. OPEN Is there a sequence$1\leq d_1<d_2<\cdots$with density$1$such that all products$\prod_{u\leq i\leq v}d_i$are distinct? OPEN Let$f(1)=f(2)=1$and for$n>2$$f(n) = f(n-f(n-1))+f(n-f(n-2)).$ Does$f(n)$miss infinitely many integers? What is its behaviour? Asked by Hofstadter. The sequence begins$1,1,2,3,3,4,\ldots$and is A005185 in the OEIS. It is not even known whether$f(n)$is well-defined for all$n$. OPEN Let$a_1=1$and$a_2=2$and for$k\geq 3$we choose$a_k$to be the least integer$>a_{k-1}$which is the sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence? Asked by Hofstadter. The sequence begins$1,2,3,5,6,8,10,11,\ldots$and is A005243 in the OEIS. OPEN Let$a_1=2$and$a_2=3$and continue the sequence by appending to$a_1,\ldots,a_n$all possible values of$a_ia_j-1$with$i\neq j$. Is it true that the set of integers which eventually appear has positive density? Asked by Hofstadter. The sequence begins$2,3,5,9,14,17,26,\ldots$and is A005244 in the OEIS. This problem is also discussed in section E31 of Guy's book Unsolved Problems in Number Theory. In [ErGr80] (and in Guy's book) this problem as written is asking for whether almost all integers appear in this sequence, but the answer to this is trivially no (as pointed out to me by Steinerberger): no integer$\equiv 1\pmod{3}$is ever in the sequence, so the set of integers which appear has density at most$2/3$. This is easily seen by induction, and the fact that if$a,b\in \{0,2\}\pmod{3}$then$ab-1\in \{0,2\}\pmod{3}$. Presumably it is the weaker question of whether a positive density of integers appear (as correctly asked in [Er77c]) that was also intended in [ErGr80]. Additional thanks to: Mehtaab Sawhney, Stefan Steinerberger OPEN Let$F(n)$be the maximum possible size of a subset$A\subseteq\{1,\ldots,N\}$such that the products$ab$are distinct for all$a<b$. Is there a constant$c$such that $F(n)=\pi(n)+(c+o(1))n^{3/4}(\log n)^{-3/2}?$ If$A\subseteq \{1,\ldots,n\}$is such that all products$a_1\cdots a_r$are distinct for$a_1<\cdots <a_r$then is it true that $\lvert A\rvert \leq \pi(n)+O(n^{\frac{r+1}{2r}})?$ Erdős [Er68] proved that there exist some constants$0<c_1\leq c_2$such that $\pi(n)+c_1 n^{3/4}(\log n)^{-3/2}\leq F(n)\leq \pi(n)+c_2 n^{3/4}(\log n)^{-3/2}.$ Surprisingly, if we consider the corresponding problem in the reals (so consider the largest$A\subset [1,x]$such that for any distinct$a,b,c,d\in A$we have$\lvert ab-cd\rvert \geq 1$) then Alexander proved that$\lvert A\rvert> x/8e$is possible (disproving an earlier conjecture of Erdős [Er73] that$m=o(x)$). Alexander's construction seems to be unpublished, and I have no idea what it is. See also [490], [793], and [796]. Additional thanks to: Rishika Agrawal OPEN -$100
We say $H$ is a unique subgraph of $G$ if there is exactly one way to find $H$ as a subgraph (not necessarily induced) of $G$. Is there a graph on $n$ vertices with $\gg \frac{2^{\binom{n}{2}}}{n!}$ many distinct unique subgraphs?
A problem of Erdős and Entringer [EnEr72], who constructed a graph with $\gg 2^{\binom{n}{2}-O(n^{3/2+o(1)})}$ many unique subgraphs. This was improved by Harary and Schwenk [HaSc73] and then by Brouwer [Br75], who constructed a graph with $\gg \frac{2^{\binom{n}{2}-O(n)}}{n!}$ many unique subgraphs.

Note that there are $\sim 2^{\binom{n}{2}}/n!$ many non-isomorphic graphs on $n$ vertices (folklore, often attributed to Pólya), and hence the bound in the problem statement is trivially best possible.

Erdős believed Brouwer's construction was essentially best possible, but Spencer suggested that $\gg \frac{2^{\binom{n}{2}}}{n!}$ may be possible. Erdős offered \$100 for a construction and \$25 for a proof that no such construction is possible.

SOLVED
Is it true that, for every $n$ and $d$, there exists $k$ such that $d \mid p_{n+1}+\cdots+p_{n+k},$ where $p_r$ denotes the $r$th prime?
Cedric Pilatte has observed that a positive solution to this follows from a result of Shiu [Sh00]: for any $k\geq 1$ and $(a,q)=1$ there exist infinitely many $k$-tuples of consecutive primes $p_m,\ldots,p_{m+{k-1}}$ all of which are congruent to $a$ modulo $q$.

Indeed, we apply this with $k=q=d$ and $a=1$ and let $p_m,\ldots,p_{m+{d-1}}$ be consecutive primes all congruent to $1$ modulo $d$, with $m>n+1$. If $p_{n+1}+\cdots+p_{m-1}\equiv r\pmod{d}$ with $1\leq r\leq d$ then $d \mid p_{n+1}+\cdots +p_m+\cdots+p_{m+r-1}.$

OPEN
Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0<a<n$ and $\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$
Erdős and Graham could show this is true (assuming the prime $k$-tuple conjecture) if we replace $\liminf$ by $\limsup$.
SOLVED
Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists some $n$ such that $n+a$ is prime for all $a\in A$?
Weisenberg [We24] has shown the answer is no: $A$ can be arbitrarily sparse and missing at least one residue class modulo every prime $p$, and yet $A+n$ is not contained in the primes for any $n\in \mathbb{Z}$.
OPEN
Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ such that all of the prime factors of $a_k$ are $>n-a_k$. Is it true that, for sufficiently large $n$, not all of this sequence can be prime?
Erdős and Graham write 'preliminary calculations made by Selfridge indicate that this is the case but no proof is in sight'. For example if $n=8$ we have $a_1=7$ and $a_2=5$ and then must stop.
OPEN
Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?
A problem of Ostmann, sometimes known as the 'inverse Goldbach problem'. The answer is surely no. The best result in this direction is due to Elsholtz and Harper [ElHa15], who showed that if $A,B$ are such sets then for all large $x$ we must have $\frac{x^{1/2}}{\log x\log\log x} \ll \lvert A \cap [1,x]\rvert \ll x^{1/2}\log\log x$ and similarly for $B$.

Elsholtz [El01] has proved there are no infinite sets $A,B,C$ such that $A+B+C$ agrees with the set of prime numbers up to finitely many exceptions.

OPEN
Let $A,B\subseteq \mathbb{N}$ be two infinite sets. How dense can $A+B$ be if all elements of $A+B$ are pairwise relatively prime?
Asked by Straus, inspired by a problem of Ostmann (see [431]).
OPEN
If $A\subset \mathbb{N}$ is a finite set then let $G(A)$ denote the greatest integer which is not expressible as a finite sum of elements from $A$ (with repetitions allowed). Let $g(n,t)=\max G(A)$ where the maximum is taken over all $A\subseteq \{1,\ldots,t\}$ of size $\lvert A\rvert=n$ which has no common divisor. Is it true that $g(n,t)\sim \frac{t^2}{n-1}?$
This type of problem is associated with Frobenius. Erdős and Graham [ErGr72] proved $g(n,t)<2t^2/n$, and there are examples which show that $g(n,t) \geq \frac{t^2}{n-1}-5t$ for $n\geq 2$.

The problem is written as Erdős and Graham describe it, but presumably they had in mind the regime where $n$ is fixed and $t\to \infty$.

OPEN
Let $k\leq n$. What choice of $A\subseteq \{1,\ldots,n\}$ of size $\lvert A\rvert=k$ maximises the number of integers not representable as the sum of finitely many elements from $A$ (with repetitions allowed)? Is it $\{n,n-1,\ldots,n-k+1\}$?
Associated with problems of Frobenius.
OPEN
Let $n\in\mathbb{N}$ with $n\neq p^k$ for any prime $p$ and $k\geq 0$. What is the largest integer not of the form $\sum_{1\leq i<n}c_i\binom{n}{i}$ where the $c_i\geq 0$ are integers?
OPEN
If $p$ is a prime and $k,m\geq 2$ then let $r(k,m,p)$ be the minimal $r$ such that $r,r+1,\ldots,r+m-1$ are all $k$th power residues modulo $p$. Let $\Lambda(k,m)=\limsup_{p\to \infty} r(k,m,p).$ Is it true that $\Lambda(k,2)$ is finite for all $k$? Is $\Lambda(k,3)$ finite for all odd $k$? How large are they?
Asked by Lehmer and Lehmer [LeLe62]. For example $\Lambda(2,2)=9$ and $\Lambda(3,2)=77$. It is known that $\Lambda(k,3)=\infty$ for all even $k$ and $\Lambda(k,4)=\infty$ for all $k$.

This has been partially resolved: Hildebrand [Hi91] has shown that $\Lambda(k,2)$ is finite for all $k$.

OPEN
Let $1\leq a_1<\cdots<a_k\leq x$. How many of the partial products $a_1,a_1a_2,\ldots,a_1\cdots a_k$ can be squares? Is it true that, for any $\epsilon>0$, there can be more than $x^{1-\epsilon}$ squares?
There are trivially $o(x)$ many squares.
SOLVED
How large can $A\subseteq \{1,\ldots,N\}$ be if $A+A$ contains no square numbers?
Taking all integers $\equiv 1\pmod{3}$ shows that $\lvert A\rvert\geq N/3$ is possible. This can be improved to $\tfrac{11}{32}N$ by taking all integers $\equiv 1,5,9,13,14,17,21,25,26,29,30\pmod{32}$, as observed by Massias.

Lagarias, Odlyzko, and Shearer [LOS83] proved this is sharp for the modular version of the problem; that is, if $A\subseteq \mathbb{Z}/N\mathbb{Z}$ is such that $A+A$ contains no squares then $\lvert A\rvert\leq \tfrac{11}{32}N$. They also prove the general upper bound of $\lvert A\rvert\leq 0.475N$ for the integer problem.

In fact $\frac{11}{32}$ is sharp in general, as shown by Khalfalah, Lodha, and Szemerédi [KLS02], who proved that the maximal such $A$ satisfies $\lvert A\rvert\leq (\tfrac{11}{32}+o(1))N$.

SOLVED
Is it true that, in any finite colouring of the integers, there must be two integers $x\neq y$ of the same colour such that $x+y$ is a square? What about a $k$th power?
A question of Roth, Erdős, Sárközy, and Sós. In other words, if $G$ is the infinite graph on $\mathbb{N}$ where we connect $m,n$ by an edge if and only if $n+m$ is a square, then is the chromatic number of $G$ equal to $\aleph_0$?

This is true, as proved by Khalfalah and Szemerédi [KhSz06], who in fact prove the general result with $x+y=z^2$ replaced by $x+y=f(z)$ for any non-constant $f(z)\in \mathbb{Z}[z]$ such that $2\mid f(z)$ for some $z\in \mathbb{Z}$.

OPEN
Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ be infinite and let $A(x)$ count the numer of indices for which $\mathrm{lcm}(a_i,a_{i+1})\leq x$. Is it true that $A(x) \ll x^{1/2}$? How large can $\liminf \frac{A(x)}{x^{1/2}}$ be?
It is easy to give a sequence with $\limsup\frac{A(x)}{x^{1/2}}=c>0.$ There are related results (particularly for the more general case of $\mathrm{lcm}(a_i,a_{i+1},\ldots,a_{i+k})$) in a paper of Erdős and Szemerédi [ErSz80].
SOLVED
Let $N\geq 1$. What is the size of the largest $A\subset \{1,\ldots,N\}$ such that $\mathrm{lcm}(a,b)\leq N$ for all $a,b\in A$?

Is it attained by choosing all integers in $[1,(N/2)^{1/2}]$ together with all even integers in $[(N/2)^{1/2},(2N)^{1/2}]$?

Let $g(N)$ denote the size of the largest such $A$. The construction mentioned proves that $g(N) \geq \left(\tfrac{9}{8}n\right)^{1/2}+O(1).$ Erdős [Er51b] proved $g(N) \leq (4n)^{1/2}+O(1)$. Chen [Ch98] established the asymptotic $g(N) \sim \left(\tfrac{9}{8}n\right)^{1/2}.$ Chen and Dai [DaCh06] proved that $g(N)\leq \left(\tfrac{9}{8}n\right)^{1/2}+O\left(\left(\frac{N}{\log N}\right)^{1/2}\log\log N\right).$ In [ChDa07] the same authors prove that, infinitely often, Erdős' construction is not optimal: if $B$ is that construction and $A$ is such that $\lvert A\rvert=g(N)$ then, for infinitely many $N$, $\lvert A\rvert\geq \lvert B\rvert+t,$ where $t\geq 0$ is defined such that the $t$-fold iterated logarithm of $N$ is in $[0,1)$.
SOLVED
Is it true that if $A\subseteq\mathbb{N}$ is such that $\frac{1}{\log\log x}\sum_{n\in A\cap [1,x)}\frac{1}{n}\to \infty$ then $\left(\sum_{n\in A\cap [1,x)}\frac{1}{n}\right)^{-2} \sum_{\substack{a,b\in A\cap (1,x]\\ a<b}}\frac{1}{\mathrm{lcm}(a,b)}\to \infty?$
Tao [Ta24b] has shown this is false: there exists $A\subset\mathbb{N}$ such that $\sum_{n\in A\cap [1,x)}\gg \exp((\tfrac{1}{2}+o(1))\sqrt{\log\log x}\log\log\log x)$ and $\left(\sum_{n\in A\cap [1,x)}\frac{1}{n}\right)^{-2} \sum_{\substack{a,b\in A\cap (1,x]\\ a<b}}\frac{1}{\mathrm{lcm}(a,b)}\ll 1.$ Moreover, Tao shows this is the best possible result, in that if $\sum_{n\in A\cap [1,x)}\frac{1}{n}$ grows faster than $\exp(O(\sqrt{\log\log x}\log\log\log x))$ then $\left(\sum_{n\in A\cap [1,x)}\frac{1}{n}\right)^{-2} \sum_{\substack{a,b\in A\cap (1,x]\\ a<b}}\frac{1}{\mathrm{lcm}(a,b)}\to \infty.$
OPEN
Let $m,n\geq 1$. What is $\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}?$ Can it be arbitrarily large? Is it $\leq (mn)^{o(1)}$ for all suffiicently large $m,n$?
SOLVED
Let $A\subseteq\mathbb{N}$ be infinite and $d_A(n)$ count the number of $a\in A$ which divide $n$. Is it true that, for every $k$, $\limsup_{x\to \infty} \frac{\max_{n<x}d_A(n)}{\left(\sum_{n\in A\cap[1,x)}\frac{1}{n}\right)^k}=\infty?$
The answer is yes, proved by Erdős and Sárkőzy [ErSa80].
OPEN
Is it true that, for any $c>1/2$, if $p$ is a large prime and $n$ is sufficiently large (both depending on $c$) then there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$?
An unpublished result of Heilbronn guarantees this for $c$ sufficiently close to $1$.
SOLVED
Let $\delta(n)$ denote the density of integers which are divisible by some integer in $(n,2n)$. What is the growth rate of $\delta(n)$?

If $\delta'(n)$ is the density of integers which have exactly one divisor in $(n,2n)$ then is it true that $\delta'(n)=o(\delta(n))$?

Besicovitch [Be34] proved that $\liminf \delta(n)=0$. Erdős [Er35] proved that $\delta(n)=o(1)$. Erdős [Er60] proved that $\delta(n)=(\log n)^{-\alpha+o(1)}$ where $\alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\cdots.$ This estimate was refined by Tenenbaum [Te84], and the true growth rate of $\delta(n)$ was determined by Ford [Fo08] who proved $\delta(n)\asymp \frac{1}{(\log n)^\alpha(\log\log n)^{3/2}}.$

Among many other results in [Fo08], Ford also proves that the second conjecture is false, and more generally that if $\delta_r(n)$ is the density of integers with exactly $r$ divisors in $(n,2n)$ then $\delta_r(n)\gg_r\delta(n)$.

Additional thanks to: Zachary Chase and Kevin Ford
SOLVED
How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be? By union-free we mean there are no solutions to $A\cup B=C$ with distinct $A,B,C\in \mathcal{F}$. Must $\lvert \mathcal{F}\rvert =o(2^n)$? Perhaps even $\lvert \mathcal{F}\rvert <(1+o(1))\binom{n}{\lfloor n/2\rfloor}?$
The estimate $\lvert \mathcal{F}\rvert=o(2^n)$ implies that if $A\subset \mathbb{N}$ is a set of positive density then there are infinitely many distinct $a,b,c\in A$ such that $[a,b]=c$ (i.e. [487]).

Solved by Kleitman [Kl71], who proved $\lvert \mathcal{F}\rvert <(1+o(1))\binom{n}{\lfloor n/2\rfloor}.$

SOLVED
Let $\tau(n)$ count the divisors of $n$ and $\tau^+(n)$ count the number of $k$ such that $n$ has a divisor in $[2^k,2^{k+1})$. Is it true that, for all $\epsilon>0$, $\tau^+(n) < \epsilon \tau(n)$ for almost all $n$?
This is false, and was disproved by Erdős and Tenenbaum [ErTe81], who showed that in fact the upper density of the set of such $n$ is $\asymp \epsilon^{1-o(1)}$ (where the $o(1)$ in the exponent $\to 0$ as $\epsilon \to 0$).

A more precise result was proved by Hall and Tenenbaum [HaTe88] (see Section 4.6), who showed that the upper density is $\ll\epsilon \log(2/\epsilon)$. Hall and Tenenbaum further prove that $\tau^+(n)/\tau(n)$ has a distribution function.

Erdős and Graham also asked whether there is a good inequality known for $\sum_{n\leq x}\tau^+(n)$. This was provided by Ford [Fo08] who proved $\sum_{n\leq x}\tau^+(n)\asymp x\frac{(\log x)^{1-\alpha}}{(\log\log x)^{3/2}}$ where $\alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\cdots.$

SOLVED
Let $r(n)$ count the number of $d_1,d_2$ such that $d_1\mid n$ and $d_2\mid n$ and $d_1<d_2<2d_1$. Is it true that, for every $\epsilon>0$, $r(n) < \epsilon \tau(n)$ for almost all $n$, where $\tau(n)$ is the number of divisors of $n$?
This is false - indeed, for any constant $K>0$ we have $r(n)>K\tau(n)$ for a positive density set of $n$. Kevin Ford has observed this follows from the negative solution to [448]: the Cauchy-Schwarz inequality implies $r(n)+\tau(n)\geq \tau(n)^2/\tau^+(n)$ where $\tau^+(n)$ is as defined in [448], and the negative solution to [448] implies the right-hand side is at least $(K+1)\tau(n)$ for a positive density set of $n$. (This argument is given for an essentially identical problem by Hall and Tenenbaum [HaTe88], Section 4.6.)

OPEN
How large must $y=y(\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\epsilon y$?
OPEN
Estimate $n_k$, the smallest integer such that $\prod_{1\leq i\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.
Erdős and Graham write 'we can prove $n_k>k^{1+c}$ but no doubt much more is true'.

In [Er79d] Erdős writes that probably $n_k<e^{o(k)}$ but $n_k>k^d$ for all constant $d$.

OPEN
Let $\omega(n)$ count the number of distinct prime factors of $n$. What is the size of the largest interval $I\subseteq [x,2x]$ such that $\omega(n)>\log\log n$ for all $n\in I$?
Erdős [Er37] proved that the density of integers $n$ with $\omega(n)>\log\log n$ is $1/2$. The Chinese remainder theorem implies that there is such an interval with $\lvert I\rvert \geq (1+o(1))\frac{\log x}{(\log\log x)^2}.$ It could be true that there is such an interval of length $(\log x)^{k}$ for arbitrarily large $k$.
SOLVED
Is it true that, for all sufficiently large $n$, $p_n^2 \leq p_{n+i}p_{n-i}$ for all $i<n$, where $p_k$ is the $k$th prime?
Asked by Erdős and Straus. The answer is no, as shown by Pomerance [Po79].
OPEN
Let $f(n) = \min_{i<n} (p_{n+i}+p_{n-i}),$ where $p_k$ is the $k$th prime. Is it true that $\limsup_n (f(n)-2p_n)=\infty?$
Pomerance [Po79] has proved the $\limsup$ is at least $2$.
OPEN
Let $q_1<q_2<\cdots$ be a sequence of primes such that $q_{n+1}-q_n\geq q_n-q_{n-1}.$ Must $\lim_n \frac{q_n}{n^2}=\infty?$
Richter [Ri76] proved that $\liminf_n \frac{q_n}{n^2}>2.84\cdots.$
OPEN
Let $p_n$ be the smallest prime $\equiv 1\pmod{n}$ and let $m_n$ be the smallest integer such that $n\mid \phi(m_n)$. Is it true that $p_n>m_n$ for almost all $n$? Does $p_n/m_n\to \infty$ for almost all $n$? Are there infinitely many primes $p$ such that $p-1$ is the only $n$ for which $m_n=p$?
Linnik's theorem implies that $p_n\leq n^{O(1)}$. Erdős [Er79e] writes it is 'easy to show' that for infinitely many $n$ we have $p_n <m_n$.
OPEN
Is there some $\epsilon>0$ such that there are infinitely many $n$ where all primes $p\leq (2+\epsilon)\log n$ divide $\prod_{1\leq i\leq \log n}(n+i)?$
A problem of Erdős and Pomerance.

More generally, let $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. This problem asks whether $q(n,\log n)\geq (2+\epsilon)\log n$ infinitely often. Taking $n$ to be the product of primes between $\log n$ and $(2+o(1))\log n$ gives an example where $q(n,\log n)\geq (2+o(1))\log n.$

Can one prove that $q(n,\log n)<(1-\epsilon)(\log n)^2$ for all large $n$ and some $\epsilon>0$?

OPEN
Let $[1,\ldots,n]$ denote the least common multiple of $\{1,\ldots,n\}$. Is it true that, for all $k\geq 1$, $[1,\ldots,p_{k+1}-1]< p_k[1,\ldots,p_k]?$
Erdős and Graham write this is 'almost certainly' true, but the proof is beyond our ability, for two reasons (at least):
• Firstly, one has to rule out the possibility of many primes $q$ such that $p_k<q^2<p_{k+1}$. There should be at most one such $q$, which would follow from $p_{k+1}-p_k<p_k^{1/2}$, which is essentially the notorious Legendre's conjecture.
• The small primes also cause trouble.
OPEN
Let $f(u)$ be the largest $v$ such that no $m\in (u,v)$ is composed entirely of primes dividing $uv$. Estimate $f(u)$.
OPEN
Let $a_0=n$ and $a_1=1$, and in general $a_k$ is the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $1\leq i<k$. Does $\sum_{i}\frac{1}{a_i}\to \infty$ as $n\to \infty$? What about if we restrict the sum to those $i$ such that $n-a_j$ is divisible by some prime $\leq a_j$, or the complement of such $i$?
This question arose in work of Eggleton, Erdős, and Selfridge.
OPEN
Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p<t$. Let $f(n,t)$ count the number of distinct possible values for $s_t(m)$ for $m\in [n+1,n+t]$. Is there an $\epsilon>0$ such that $f(n,t)>\epsilon t$ for all $t$ and $n$?
Erdős and Graham report they can show $f(n,t) \gg \frac{t}{\log t}.$
OPEN
Let $p(n)$ denote the least prime factor of $n$. There is a constant $c>0$ such that $\sum_{\substack{n<x\\ n\textrm{ not prime}}}\frac{p(n)}{n}\sim c\frac{x^{1/2}}{(\log x)^2}.$ Is it true that there exists a constant $C>0$ such that $\sum_{x\leq n\leq x+Cx^{1/2}(\log x)^2}\frac{p(n)}{n} \gg 1$ for all large $x$?
OPEN
Is there a function $f$ with $f(n)\to \infty$ as $n\to \infty$ such that, for all large $n$, there is a composite number $m$ such that $n+f(n)<m<n+p(m)?$ (Here $p(m)$ is the least prime factor of $m$.)
SOLVED
Let $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ be a lacunary sequence (so there exists some $\lambda>1$ with $a_{k+1}\geq \lambda a_k$ for all $k$). Is there an irrational $\alpha$ such that $\{ \{\alpha a_k\} : k\geq 1\}$ is not everywhere dense in $[0,1]$ (where $\{x\}=x-\lfloor x\rfloor$ is the fractional part).
Erdős and Graham write the existence of such an $\alpha$ has 'very recently been shown', but frustratingly give neither a name nor a reference. I will try to track down this solution soon.
SOLVED
Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that $\| \lvert P_i-P_j\rvert \| \geq \delta$ for all $1\leq i<j\leq n$. (Here $\|x\|$ is the distance from $x$ to the nearest integer.)

Is it true that, for any $0<\delta<1/2$, we have $N(X,\delta)=o(X)?$ In fact, is it true that (for any fixed $\delta>0$) $N(X,\delta)<X^{1/2+o(1)}?$

The first conjecture was proved by Sárközy [Sa76], who in fact proved $N(X,\delta) \ll \delta^{-3}\frac{X}{\log\log X}.$ See also [466].

Even a stronger statement is true, as shown by Konyagin [Ko01], who proved that $N(X,\delta) \ll_\delta N^{1/2}.$

SOLVED
Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that $\| \lvert P_i-P_j\rvert \| \geq \delta$ for all $1\leq i<j\leq n$. (Here $\|x\|$ is the distance from $x$ to the nearest integer.)

Is there some $\delta>0$ such that $\lim_{x\to \infty}N(X,\delta)=\infty?$

Graham proved this is true, and in fact $N(X,1/10)> \frac{\log X}{10}.$ This was substantially improved by Sárközy [Sa76], who proved that for, all sufficiently small $\delta>0$, $N(X,\delta)>X^{1/2-\delta^{1/7}}.$ See also [465].
OPEN
Prove the following for all large $x$: there is a choice of congruence classes $a_p$ for all primes $p\leq x$ and a decomposition $\{p\leq x\}=A\sqcup B$ into two non-empty sets such that, for all $n<x$, there exist some $p\in A$ and $q\in B$ such that $n\equiv a_p\pmod{p}$ and $n\equiv a_q\pmod{q}$.
This is what I assume the intended problem is, although the presentation in [ErGr80] is missing some crucial quantifiers, so I may have misinterpreted it.
OPEN
For any $n$ let $D_n$ be the set of sums of the shape $d_1,d_1+d_2,d_1+d_2+d_3,\ldots$ where $1<d_1<d_2<\cdots$ are the divisors of $n$.

What is the size of $D_n\backslash \cup_{m<n}D_m$?

If $f(N)$ is the minimal $n$ such that $N\in D_n$ then is it true that $f(N)=o(N)$? Perhaps just for almost all $N$?

OPEN
Let $A$ be the set of all $n$ such that $n=d_1+\cdots+d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m\mid n$ with $m<n$. Does $\sum_{n\in A}\frac{1}{n}$ converge?
The same question can be asked for those $n$ which do not have distinct sums of sets of divisors, but any proper divisor of $n$ does.
OPEN
Call $n$ weird if $\sigma(n)\geq 2n$ and $n\neq d_1+\cdots+d_k$, where the $d_i$ are distinct proper divisors of $n$. Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird?
Weird numbers were investigated by Benkoski and Erdős [BeEr74], who proved that the set of weird numbers has positive density.

Melfi [Me15] has proved that there are infinitely many primitive weird numbers, conditional on various well-known conjectures on the distribution of prime gaps. For example, it would suffice to show that $p_{n+1}-p_n <\frac{1}{10}p_n^{1/2}$ for sufficiently large $n$.

OPEN
Given a finite set of primes $Q=Q_0$, define a sequence of sets $Q_i$ by letting $Q_{i+1}$ be $Q_i$ together with all primes formed by adding three distinct elements of $Q_i$. Is there some initial choice of $Q$ such that the $Q_i$ become arbitrarily large?
A problem of Ulam. In particular, what about $Q=\{3,5,7,11\}$?
OPEN
Given some initial finite sequence of primes $q_1<\cdots<q_m$ extend it so that $q_{n+1}$ is the smallest prime of the form $q_n+q_i-1$ for $n\geq m$. Is there an initial starting sequence so that the resulting sequence is infinite?
A problem due to Ulam. For example if we begin with $3,5$ then the sequence continues $3,5,7,11,13,17,\ldots$. It is possible that this sequence is infinite.
SOLVED
Is there a permutation $a_1,a_2,\ldots$ of the positive integers such that $a_k+a_{k+1}$ is always prime?

Watts has suggested that perhaps the obvious greedy algorithm defines such a permutation - that is, let $a_1=1$ and let $a_{n+1}=\min \{ x : a_n+x\textrm{ is prime and }x\neq a_i\textrm{ for }i\leq n\}.$ In other words, do all positive integers occur as some such $a_n$? Do all primes occur as a sum?

OPEN
Let $b_1=1$ and in general let $b_{n+1}$ be the least integer which is not of the shape $\sum_{u\leq i\leq v}b_i$ for some $1\leq u\leq v\leq n$. How does this sequence grow?
The sequence is OEIS A002048 and begins $1,2,4,5,8,10,14,15,16,21,22,23,25,\ldots.$ In general, if $a_1<a_2<\cdots$ is a sequence so that no $a_n$ is a sum of consecutive $a_i$s, then must the density of the $a_i$ be zero? What about the lower density?
OPEN
Let $p$ be a prime. Given any finite set $A\subseteq \mathbb{F}_p\backslash \{0\}$, is there always a rearrangement $A=\{a_1,\ldots,a_t\}$ such that all partial sums $\sum_{1\leq k\leq m}a_{k}$ are distinct, for all $1\leq m\leq t$?
A problem of Graham, who proved it when $t=p-1$. A similar conjecture was made for arbitrary abelian groups by Alspach. Such an ordering is often called a valid ordering.

This has been proved for $t\leq 12$ (see Costa and Pellegrini [CoPe20] and the references therein) and for $p-3\leq t\leq p-1$ (see Hicks, Ollis, and Schmitt [HOS19] and the references therein). Kravitz [Kr24] has proved this for $t \leq \frac{\log p}{\log\log p}.$

SOLVED
Let $A\subseteq \mathbb{F}_p$. Let $A\hat{+}A = \{ a+b : a\neq b \in A\}.$ Is it true that $\lvert A\hat{+}A\rvert \geq \min(2\lvert A\rvert-3,p)?$
A question of Erdős and Heilbronn. Solved in the affirmative by da Silva and Hamidoune [dSHa94].
OPEN
Let $f:\mathbb{Z}\to \mathbb{Z}$ be a polynomial of degree at least $2$. Is there a set $A$ such that every $z\in \mathbb{Z}$ has exactly one representation as $z=a+f(n)$ for some $a\in A$ and $n\in \mathbb{Z}$?
Probably there is no such $A$ for any polynomial $f$.
OPEN
Let $p$ be a prime and $A_p = \{ k! \mod{p} : 1\leq k<p\}.$ Is it true that $\lvert A_p\rvert \sim (1-\tfrac{1}{e})p?$
OPEN
Is it true that, for all $k\neq 1$, there are infinitely many $n$ such that $2^n\equiv k\pmod{n}$?
A conjecture of Graham. It is easy to see that $2^n\not\equiv 1\mod{n}$ for all $n>1$, so the restriction $k\neq 1$ is necessary. Erdős and Graham report that Graham, Lehmer, and Lehmer have proved this for $k=2^i$ for $i\geq 1$, or if $k=-1$, but I cannot find such a paper.

As an indication of the difficulty, when $k=3$ the smallest $n$ such that $2^n\equiv 3\pmod{n}$ is $n=4700063497$.

SOLVED
Let $x_1,x_2,\ldots\in [0,1]$ be an infinite sequence. Is it true that there are infinitely many $m,n$ such that $\lvert x_{m+n}-x_n\rvert \leq \frac{1}{\sqrt{5}n}?$
A conjecture of Newman. This was proved Chung and Graham, who in fact show that for any $\epsilon>0$ there must exist some $n$ such that there are infinitely many $m$ for which $\lvert x_{m+n}-x_m\rvert < \frac{1}{(c-\epsilon)n}$ where $c=1+\sum_{k\geq 1}\frac{1}{F_{2k}}=2.535\cdots$ and $F_m$ is the $m$th Fibonacci number. This constant is best possible.
OPEN
Let $a_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N}$ such that $\sum_{i}\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\ldots,x_n)$ let $T(A)$ denote the sequence of length $rn$ given by $(a_ix_j+b_i)_{1\leq j\leq n, 1\leq i\leq r}.$ Prove that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated elements.
Erdős and Graham write that 'it is surprising that [this problem] offers difficulty'.

The original formulation of this problem had an extra condition on the minimal element of the sequence $A_k$ being large, but Ryan Alweiss has pointed out that is trivially always satisfied since the minimal element of the sequence must grow by at least $1$ at each stage.

Additional thanks to: Ryan Alweiss, Zachary Chase
SOLVED
Define a sequence by $a_1=1$ and $a_{n+1}=\lfloor\sqrt{2}(a_n+1/2)\rfloor$ for $n\geq 1$. The difference $a_{2n+1}-2a_{2n-1}$ is the $n$th digit in the binary expansion of $\sqrt{2}$.

Find similar results for $\theta=\sqrt{m}$, and other algebraic numbers.

The result for $\sqrt{2}$ was obtained by Graham and Pollak [GrPo70]. The problem statement is open-ended, but presumably Erdős and Graham would have been satisfied with the wide-ranging generalisations of Stoll ([St05] and [St06]).
OPEN
Let $f(k)$ be the minimal $N$ such that if $\{1,\ldots,N\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $f(k)$. In particular, is it true that $f(k) < c^k$ for some constant $c>0$?
Schur proved that $f(k)<ek!$. See also [183].
OPEN
Prove that there exists an absolute constant $c>0$ such that, whenever $\{1,\ldots,N\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\{1,\ldots,N\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\in \{1,\ldots,N\}$ are in the same colour class).
A conjecture of Roth.
SOLVED
Let $f(k)$ be the minimum number of terms in $P(x)^2$, where $P\in \mathbb{Q}[x]$ ranges over all polynomials with exactly $k$ non-zero terms. Is it true that $f(k)\to\infty$ as $k\to \infty$?
First investigated by Rényi and Rédei [Re47]. Erdős [Er49b] proved that $f(k)<k^{1-c}$ for some $c>0$. The conjecture that $f(k)\to \infty$ is due to Erdős and Rényi.

This was solved by Schinzel [Sc87], who proved that $f(k) > \frac{\log\log k}{\log 2}.$ In fact Schinzel proves lower bounds for the corresponding problem with $P(x)^n$ for any integer $n\geq 1$, where the coefficients of the polynomial can be from any field with zero or sufficiently large positive characteristic.

Schinzel and Zannier [ScZa09] have improved this to $f(k) \gg \log k.$

OPEN
Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let $B = \{ m\in \mathbb{N} : m\not\in X_n\pmod{n}\textrm{ for all }n\in A\}.$ Must $B$ have a logarithmic density, i.e. is it true that $\lim_{x\to \infty} \frac{1}{\log x}\sum_{\substack{m\in B\\ m<x}}\frac{1}{m}$ exists?
Davenport and Erdős [DaEr37] proved that the answer is yes when $X_n=\{0\}$ for all $n\in A$. The problem considers logarithmic density since Besicovitch [Be34] showed examples exist without a natural density, even when $X_n=\{0\}$ for all $n\in A$.
SOLVED
Let $A\subseteq \mathbb{N}$ have positive density. Must there exist distinct $a,b,c\in A$ such that $[a,b]=c$ (where $[a,b]$ is the lowest common multiple of $a$ and $b$)?
Davenport and Erdős [DaEr37] showed that there must exist an infinite sequence $a_1<a_2\cdots$ in $A$ such that $a_i\mid a_j$ for all $i\leq j$.

This is true, a consequence of the positive solution to [447] by Kleitman [Kl71].

OPEN
Let $A$ be a finite set and $B=\{ n \geq 1 : a\nmid n\textrm{ for all }a\in A\}.$ Is it true that, for every $m>n$, $\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?$
The example $A=\{a\}$ and $n=2a-1$ and $m=2a$ shows that $2$ would be best possible.
OPEN
Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let $B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}.$ If $B=\{b_1<b_2<\cdots\}$ then is it true that $\lim \frac{1}{x}\sum_{b_i<x}(b_{i+1}-b_i)^2$ exists (and is finite)?
For example, when $A=\{p^2: p\textrm{ prime}\}$ then $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erdős.

SOLVED
Let $A,B\subseteq \{1,\ldots,N\}$ be such that all the products $ab$ with $a\in A$ and $b\in B$ are distinct. Is it true that $\lvert A\rvert \lvert B\rvert \ll \frac{N^2}{\log N}?$
This would be best possible, for example letting $A=[1,N/2]\cap \mathbb{N}$ and $B=\{ N/2<p\leq N: p\textrm{ prime}\}$.

This is true, and was proved by Szemerédi [Sz76].

SOLVED
Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). If there is a constant $c$ such that $\lvert f(n+1)-f(n)\rvert <c$ for all $n$ then must there exist some $c'$ such that $f(n)=c'\log n+O(1)?$
Erdős [Er46] proved that if $f(n+1)-f(n)=o(1)$ or $f(n+1)\geq f(n)$ then $f(n)=c\log n$ for some constant $c$.

This is true, and was proved by Wirsing [Wi70].

SOLVED
Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ be infinite such that $a_{i+1}/a_i\to 1$. For any $x\geq a_1$ let $f(x) = \frac{x-a_i}{a_{i+1}-a_i}\in [0,1),$ where $x\in [a_i,a_{i+1})$. Is it true that, for almost all $\alpha$, the sequence $f(\alpha n)$ is uniformly distributed in $[0,1)$?
For example if $A=\mathbb{N}$ then $f(x)=\{x\}$ is the usual fractional part operator.

A problem due to Le Veque [LV53], who proved it in some special cases.

This is false is general, as shown by Schmidt [Sc69].

SOLVED
Does there exist a $k$ such that every sufficiently large integer can be written in the form $\prod_{i=1}^k a_i - \sum_{i=1}^k a_i$ for some integers $a_i\geq 2$?
Erdős attributes this question to Schinzel. Eli Seamans has observed that the answer is yes (with $k=2$) for a very simple reason: $n = 2(n+2)-(2+(n+2)).$ There may well have been some additional constraint in the problem as Schinzel posed it, but [Er61] does not record what this is.
OPEN
Let $A\subset \mathbb{C}$ be a finite set of fixed size, for any $k\geq 1$ let $A_k = \{ z_1+\cdots+z_k : z_i\in A\textrm{ distinct}\}.$ For $k>2$ does the set $A_k$ (together with the size of $A$) uniquely determine the set $A$?
A problem of Selfridge and Straus [SeSt58], who prove that this is true if $k=2$ and $\lvert A\rvert \neq 2^l$ (for $l\geq 0$). On the other hand, there are examples with two distinct $A,B$ both of size $2^l$ such that $A_2=B_2$.

More generally, they prove that $A$ is uniquely determined by $A_k$ if $n$ is divisible by a prime greater than $k$. Selfridge and Straus sound more cautious than Erdős, and it may well be that for all $k>2$ there exist $A,B$ of the same size with identical $A_k=B_k$.

(In [Er61] Erdős states this problem incorrectly, replacing sums with products. This product formulation is easily seen to be false, as observed by Steinerberger: consider the case $k=3$ and subsets of the 6th roots of unity corresponding to $\{0,1,2,4\}$ and $\{0,2,3,4\}$ (as subsets of $\mathbb{Z}/6\mathbb{Z}$). The correct problem statement can be found in the paper of Selfridge and Straus that Erdős cites.)

OPEN
Let $\alpha,\beta \in \mathbb{R}$. Is it true that $\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0$ where $\|x\|$ is the distance from $x$ to the nearest integer?
The infamous Littlewood conjecture.
SOLVED
Let $\alpha \in \mathbb{R}$ be irrational and $\epsilon>0$. Are there positive integers $x,y,z$ such that $\lvert x^2+y^2-z^2\alpha\rvert <\epsilon?$
Originally a conjecture due to Oppenheim. Davenport and Heilbronn [DaHe46] solve the analogous problem for quadratic forms in 5 variables.

This is true, and was proved by Margulis [Ma89].

SOLVED
How many antichains in $[n]$ are there? That is, how many families of subsets of $[n]$ are there such that, if $\mathcal{F}$ is such a family and $A,B\in \mathcal{F}$, then $A\not\subseteq B$?
Sperner's theorem states that $\lvert \mathcal{F}\rvert \leq \binom{n}{\lfloor n/2\rfloor}$. This is also known as Dedekind's problem.

Resolved by Kleitman [Kl69], who proved that the number of such families is $2^{(1+o(1))\binom{n}{\lfloor n/2\rfloor}}.$

SOLVED
Let $z_1,\ldots,z_n\in\mathbb{C}$. Let $D$ be an arbitrary disc of radius $1$. Is it true that the number of sums of the shape $\sum_{i=1}^n\epsilon_iz_i \textrm{ for }\epsilon_i\in \{-1,1\}$ which lie in $D$ is at most $\binom{n}{\lfloor n/2\rfloor}$?
A strong form of the Littlewood-Offord problem. Erdős proved this is true if $z_i\in\mathbb{R}$, and for general $z_i\in\mathbb{C}$ proved a weaker upper bound of $\ll \frac{2^n}{\sqrt{n}}.$ This was solved in the affirmative by Kleitman [Kl65], who also later generalised this to arbitrary Hilbert spaces [Kl70].
SOLVED
Let $M=(a_{ij})$ be a real $n\times n$ doubly stochastic matrix (i.e. the entries are non-negative and each column and row sums to $1$). Does there exist some $\sigma\in S_n$ such that $\prod_{1\leq i\leq n}a_{i\sigma(i)}\geq n^{-n}?$
A weaker form of the conjecture of van der Waerden, which states that $\mathrm{perm}(M)=\sum_{\sigma\in S_n}\prod_{1\leq i\leq n}a_{i\sigma(i)}\geq n^{-n}n!$ with equality if and only if $a_{ij}=1/n$ for all $i,j$.

This conjecture is true, and was proved by Marcus and Minc [MaMi62].

Erdős also conjectured the even weaker fact that there exists some $\sigma\in S_n$ such that $a_{i\sigma(i)}\neq 0$ for all $i$ and $\sum_{i}a_{i\sigma(i)}\geq 1.$ This weaker statement was proved by Marcus and Ree [MaRe59].

van der Waerden's conjecture itself was proved by Gyires [Gy80], Egorychev [Eg81], and Falikman [Fa81].

OPEN - $500 What is$\mathrm{ex}_3(n,K_4^3)$? That is, the largest number of$3$-edges which can placed on$n$vertices so that there exists no$K_4^3$, a set of 4 vertices which is covered by all 4 possible$3$-edges. A problem of Turán. Turán observed that dividing the vertices into three equal parts$X_1,X_2,X_3$, and taking the edges to be those triples that either have exactly one vertex in each part or two vertices in$X_i$and one vertex in$X_{i+1}$(where$X_4=X_1$) shows that $\mathrm{ex}_3(n,K_4^3)\geq\left(\frac{5}{9}+o(1)\right)\binom{n}{3}.$ This is probably the truth. The current best upper bound is $\mathrm{ex}_3(n,K_4^3)\leq 0.5611666\binom{n}{3},$ due to Razborov [Ra10]. See also [712] for the general case. OPEN For every$x\in\mathbb{R}$let$A_x\subset \mathbb{R}$be a bounded set with outer measure$<1$. Must there exist an infinite independent set, that is, some infinite$X\subseteq \mathbb{R}$such that$x\not\in A_y$for all$x\neq y\in X$? If the sets$A_x$are closed and have measure$<1$, then must there exist an independent set of size$3$? Erdős and Hajnal [ErHa60] proved the existence of arbitrarily large finite independent sets (under the assumptions in the first problem). Erdős writes in [Er61] that Gladysz has proved the existence of an independent set of size$2$in the second question, but I cannot find a reference. Hechler [He72] has shown the answer to the first question is no, assuming the continuum hypothesis. SOLVED What is the size of the largest$A\subseteq \mathbb{R}^n$such that there are only two distinct distances between elements of$A$? That is, $\# \{ \lvert x-y\rvert : x\neq y\in A\} = 2.$ Asked to Erdős by Coxeter. Erdős thought he could show that$\lvert A\rvert \leq n^{O(1)}$, but later discovered a mistake in his proof, and his proof only gave$\leq \exp(n^{1-o(1)})$. Bannai, Bannai, and Stanton [BBS83] have proved that $\lvert A\rvert \leq \binom{n+2}{2}.$ A simple proof of this upper bound was given by Petrov and Pohoata [PePo21]. Shengtong Zhang has observed that a simple lower bound of$\binom{n}{2}$is given by considering all points with exactly two coordinates equal to$1$and all others equal to$0$. Additional thanks to: Ryan Alweiss, Jordan Ellenberg, Shengtong Zhang OPEN What is the size of the largest$A\subseteq \mathbb{R}^n$such that every three points from$A$determine an isosceles triangle? That is, for any three points$x,y,z$from$A$, at least two of the distances$\lvert x-y\rvert,\lvert y-z\rvert,\lvert x-z\rvert$are equal. When$n=2$the answer is$6$(due to Kelly [ErKe47]). When$n=3$the answer is$8$(due to Croft [Cr62]). The best upper bound known in general is due to Blokhuis [Bl84] who showed that $\lvert A\rvert \leq \binom{n+2}{2}.$ Alweiss has observed a lower bound of$\binom{n+1}{2}$follows from considering the subset of$\mathbb{R}^{n+1}$formed of all vectors$e_i+e_j$where$e_i,e_j$are distinct coordinate vectors. This set can be viewed as a subset of some$\mathbb{R}^n$, and is easily checked to have the required property. The fact that the truth for$n=3$is$8$suggests that neither of these bounds is the truth. Additional thanks to: Ryan Alweiss SOLVED Let$\alpha_n$be the infimum of all$0\leq \alpha\leq \pi$such that in every set$A\subset \mathbb{R}^2$of size$n$there exist three distinct points$x,y,z\in A$such that the angle determined by$xyz$is at least$\alpha$. Determine$\alpha_n$. Blumenthal's problem. Szekeres [Sz41] showed that $\alpha_{2^n+1}> \pi \left(1-\frac{1}{n}+\frac{1}{n(2^n+1)^2}\right)$ and $\alpha_{2^n}\leq \pi\left(1-\frac{1}{n}\right).$ Erdős and Szekeres [ErSz60] showed that $\alpha_{2^n}=\alpha_{2^n-1}= \pi\left(1-\frac{1}{n}\right),$ and suggested that perhaps$\alpha_{N}=\pi(1-1/n)$for$2^{n-1}<N\leq 2^n$. This was disproved by Sendov [Se92]. Sendov [Se93] provided the definitive answer, proving that$\alpha_N=\pi(1-1/n)$for$2^{n-1}+2^{n-3}<N\leq 2^n$and$\alpha_N=\pi(1-\frac{1}{2n-1})$for$2^{n-1}<N\leq 2^{n-1}+2^{n-3}$. SOLVED Is every set of diameter$1$in$\mathbb{R}^n$the union of at most$n+1$sets of diameter$<1$? Borsuk's problem. This is trivially true for$n=1$and easy for$n=2$. For$n=3$it is true, which was proved by Eggleston [Eg55]. The answer is in fact no in general, as shown by Kahn and Kalai [KaKa93], who proved that it is false for$n>2014$. The current smallest$n$where Borsuk's conjecture is known to be false is$n=64$, a result of Brouwer and Jenrich [BrJe14]. If$\alpha(n)$is the smallest number of pieces of diameter$<1$required (so Borsuk's original conjecture was that$\alpha(n)=n+1$) then Kahn and Kalai's construction shows that$\alpha(n)\geq (1.2)^{\sqrt{n}}$. The best upper bound, due to Schramm [Sc88], is that $\alpha(n) \leq ((3/2)^{1/2}+o(1))^{n}.$ OPEN What is the minimum number of circles determined by any$n$points in$\mathbb{R}^2$, not all on a circle? OPEN Let$\alpha(n)$be such that every set of$n$points in the unit circle contains three points which determine a triangle of area at most$\alpha(n)$. Estimate$\alpha(n)$. Heilbronn's triangle problem. It is trivial that$\alpha(n) \ll 1/n$. Erdős observed that$\alpha(n)\gg 1/n^2$. The current best bounds are $\frac{\log n}{n^2}\ll \alpha(n) \ll \frac{1}{n^{8/7+1/2000}}.$ The lower bound is due to Komlós, Pintz, and Szemerédi [KPS82]. The upper bound is due to Cohen, Pohoata, and Zakharov [CPZ23] (improving on an exponent of$8/7$due to Komlós, Pintz, and Szemerédi [KPS81]). OPEN What is the chromatic number of the plane? That is, what is the smallest number of colours required to colour$\mathbb{R}^2$such that no two points of the same colour are distance$1$apart? The Hadwiger-Nelson problem. Let$\chi$be the chromatic number of the plane. An equilateral triangle trivially shows that$\chi\geq 3$. There are several small graphs that show$\chi\geq 4$(in particular the Moser spindle and Golomb graph). The best bounds currently known are $5 \leq \chi \leq 7.$ The lower bound is due to de Grey [dG18]. The upper bound can be seen by colouring the plane by tesselating by hexagons with diameter slightly less than$1$. See also [704], [705], and [706]. OPEN Let$f(z)\in\mathbb{C}[z]$be a monic non-constant polynomial. Can the set $\{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\}$ be covered by a set of circles the sum of whose radii is$\leq 2$? Cartan proved this is true with$2$replaced by$2e$, which was improved to$2.59$by Pommerenke [Po61]. Pommerenke [Po59] proved that$2$is achievable if the set is connected (in fact the entire set is covered by a single circle with radius$2$). OPEN If$A\subset \mathbb{Z}$is a finite set of size$N$then is there some absolute constant$c>0$and$\theta$such that $\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?$ Chowla's cosine problem. The best known bound currently, due to Ruzsa [Ru04] (improving on an earlier result of Bourgain [Bo86]), replaces$N^{1/2}$by $\exp(O(\sqrt{\log N}).$ The example$A=B-B$, where$B$is a Sidon set, shows that$N^{1/2}$would be the best possible here. OPEN Let$f(z)\in \mathbb{C}[z]$be a monic polynomial of degree$n$and $A = \{ z\in \mathbb{C} : \lvert f(z)\rvert\leq 1\}.$ Is it true that, for every such$f$and constant$c>0$, the set$A$can have at most$O_c(1)$many components of diameter$>1+c$(where the implied constant is in particular independent of$n$)? SOLVED Is it true that, if$A\subset \mathbb{Z}$is a finite set of size$N$, then $\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N,$ where$e(x)=e^{2\pi ix }$? Littlewood's conjecture, proved independently by Konyagin [Ko81] and McGehee, Pigno, and Smith [MPS81]. OPEN Let$f=\sum_{n=0}^\infty a_nz^n$be an entire function. What is the greatest possible value of $\liminf_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}?$ It is trivial that this value is in$[1/2,1)$. Kövári (unpublished) observed that it must be$>1/2$. Clunie and Hayman [ClHa64] showed that it is$\leq 2/\pi-c$for some absolute constant$c>0$. Some other results on this quantity were established by Gray and Shah [GrSh63]. See also [227]. OPEN Let$f(z)$be an entire function. Does there exist a path$L$so that, for every$n$, $\lvert f(z)/z^n\rvert \to \infty$ as$z\to \infty$along$L$? Can the length of this path be estimated in terms of$M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$? Does there exist a path along which$\lvert f(z)\rvert$tends to$\infty$faster than a fixed function of$M(r)$(such that$M(r)^\epsilon$)? Boas (unpublished) has proved the first part, that such a path must exist. OPEN Let$f(z)$be an entire function, not a polynomial. Does there exist a path$C$such that, for every$\lambda>0$, the integral $\int_C \lvert f(z)\rvert^{-\lambda} \mathrm{d}z$ is finite? Huber [Hu57] proved that for every$\lambda>0$there is a path$C_\lambda$such that this integral is finite. OPEN Let$f(z)=\sum_{k\geq 1}a_k z^{n_k}$be an entire function of finite order such that$\lim n_k/k=\infty$. Let$M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$and$m(r)=\max_n \lvert a_nr^n\rvert$. Is it true that $\limsup\frac{\log m(r)}{\log M(r)}=1?$ A problem of Pólya. Results of Wiman [Wi14] imply that if$(n_{k+1}-n_k)^2>n_k$then$\limsup \frac{m(r)}{M(r)}=1$. Erdős and Macintyre [ErMa54] proved this under the assumption that $\sum_{k\geq 2}\frac{1}{n_{k+1}-n_k}<\infty.$ OPEN Let$f(z)=\sum_{k=1}^\infty a_kz^{n_k}$be an entire function. Is it true that if$n_k/k\to \infty$then$f(z)$assumes every value infinitely often? A conjecture of Fejér and Pólya. Fejér [Fe08] proved that if$\sum\frac{1}{n_k}<\infty$then$f(z)$assumes every value at least once, and Biernacki [Bi28] showed that this holds under the assumption that$n_k/k\to \infty$. OPEN Let$A_1,A_2,\ldots$be sets of complex numbers, none of which has a limit point in$\mathbb{C}$. Does there exist an entire function$f(z)$and a sequence$n_1<n_2<\cdots$such that$f^{(n_k)}$vanishes on$A_k$for all$k\geq 1$? SOLVED Let$z_1,\ldots,z_n\in \mathbb{C}$with$z_1=1$. Must there exist an absolute constant$c>0$such that $\max_{1\leq k\leq n}\left\lvert \sum_{i}z_i^k\right\rvert>c?$ A problem of Turán, who proved that this maximum is$\gg 1/n$. This was solved by Atkinson [At61b], who showed that$c=1/6$suffices. This has been improved by Biró, first to$c=1/2$[Bi94], and later to an absolute constant$c>1/2$[Bi00]. Based on computational evidence it is likely that the optimal value of$c$is$\approx 0.7$. SOLVED Let$f$be a Rademacher multiplicative function: a random$\{-1,0,1\}$-valued multiplicative function, where for each prime$p$we independently choose$f(p)\in \{-1,1\}$uniformly at random, and for square-free integers$n$we extend$f(p_1\cdots p_r)=f(p_1)\cdots f(p_r)$(and$f(n)=0$if$n$is not squarefree). Does there exist some constant$c>0$such that, almost surely, $\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N\log\log N}}=c?$ Note that if we drop the multiplicative assumption, and simply assign$f(m)=\pm 1$at random, then this statement is true (with$c=\sqrt{2}$), the law of the iterated logarithm. Wintner [Wi44] proved that, almost surely, $\sum_{m\leq N}f(m)\ll N^{1/2+o(1)},$ and Erdős improved the right-hand side to$N^{1/2}(\log N)^{O(1)}$. Lau, Tenenbaum, and Wu [LTW13] have shown that, almost surely, $\sum_{m\leq N}f(m)\ll N^{1/2}(\log\log N)^{2+o(1)}.$ Harper [Ha13] has shown that the sum is almost surely not$O(N^{1/2}/(\log\log N)^{5/2+o(1)})$, and conjectured that in fact Erdős' conjecture is false, and almost surely $\sum_{m\leq N}f(m) \ll N^{1/2}(\log\log N)^{1/4+o(1)}.$ This was proved by Caich [Ca23]. Additional thanks to: Mehtaab Sawhney OPEN For any$t\in (0,1)$let$t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$(where$\epsilon_k(t)\in \{0,1\}$). Let$R_n(t)$denote the number of real roots of$\sum_{1\leq k\leq n}\epsilon_k(t)z^k$. Is it true that, for almost all$t\in (0,1)$, we have $\lim_{n\to \infty}\frac{R_n(t)}{\log n}=\frac{\pi}{2}?$ Erdős and Offord [EO56] showed that the number of real roots of a random degree$n$polynomial with$\pm 1$coefficients is$(\frac{2}{\pi}+o(1))\log n$. See also [522]. OPEN Is it true that all except$o(2^n)$many polynomials of degree$n$with$\pm 1$-valued coefficients have$(\frac{1}{2}+o(1))n$many roots in$\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$? For any$t\in (0,1)$let$t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$(where$\epsilon_k(t)\in \{0,1\}$). If$S_n(t)$is the number of roots of$\sum_{1\leq k\leq n}\epsilon_k(t)z^k$in$\lvert z\rvert \leq1$then is it true that, for almost all$t\in (0,1)$, $\lim_{n\to \infty}\frac{S_n(t)}{n}=\frac{1}{2}?$ Erdős and Offord [EO56] showed that the number of real roots of a random degree$n$polynomial with$\pm 1$coefficients is$(\frac{2}{\pi}+o(1))\log n$. See also [521]. Additional thanks to: Zachary Chase OPEN For any$t\in (0,1)$let$t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$(where$\epsilon_k(t)\in \{0,1\}$). Does there exist some constant$C>0$such that, for almost all$t\in (0,1)$, $\max_{\lvert z\rvert=1}\left\lvert \sum_{k\leq n}\epsilon_k(t)z^k\right\rvert=(C+o(1))\sqrt{n\log n}?$ Salem and Zygmund [SZ54] proved that$\sqrt{n\log n}$is the right order of magnitude, but not an asymptotic. OPEN For any$t\in (0,1)$let$t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$(where$\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for almost all$t\in(0,1)$) for $M_n(t)=\max_{x\in [0,1]}\left\lvert \sum_{k\leq n}\epsilon_k(t)x^k\right\rvert?$ A problem of Salem and Zygmund [SZ54]. Chung showed that, for almost all$t$, there exist infinitely many$n$such that $M_n(t) \ll \left(\frac{n}{\log\log n}\right)^{1/2}.$ Erdős (unpublished) showed that for almost all$t$and every$\epsilon>0$we have$\lim_{n\to \infty}M_n(t)/n^{1/2-\epsilon}=\infty$. SOLVED Is it true that all except at most$o(2^n)$many degree$n$polynomials with$\pm 1$-valued coefficients$f(z)$have$\lvert f(z)\rvert <1$for some$\lvert z\rvert=1$? What is the behaviour of $m(f)=\min_{\lvert z\rvert=1}\lvert f(z)\rvert?$ Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Radamacher coefficients, i.e. independent uniform$\pm 1$values. The first problem asks whether$m(f)<1$almost surely. Littlewood [Li66] conjectured that the stronger$m(f)=o(1)$holds almost surely. The answer to both questions is yes: Littlewood's conjecture was solved by Kashin [Ka87], and Konyagin [Ko94] improved this to show that$m(f)\leq n^{-1/2+o(1)}$almost surely. This is essentially best possible, since Konyagin and Schlag [KoSc99] proved that for any$\epsilon>0$$\limsup_{n\to \infty} \mathbb{P}(m(f) \leq \epsilon n^{-1/2})\ll \epsilon.$ Cook and Nguyen [CoNg21] have identified the limiting distribution, proving that for any$\epsilon>0$$\lim_{n\to \infty} \mathbb{P}(m(f) > \epsilon n^{-1/2}) = e^{-\epsilon \lambda}$ where$\lambda$is an explicit constant. Additional thanks to: Mehtaab Sawhney SOLVED Let$a_n\geq 0$with$a_n\to 0$and$\sum a_n=\infty$. Find a necessary and sufficient condition on the$a_n$such that, if we choose (independently and uniformly) random arcs on the unit circle of length$a_n$, then all the circle is covered with probability$1$. A problem of Dvoretzky [Dv56]. It is easy to see that (under the given conditions alone) almost all the circle is covered with probability$1$. Kahane [Ka59] showed that$a_n=\frac{1+c}{n}$with$c>0$has this property, which Erd\H{s} (unpublished) improved to$a_n=\frac{1}{n}$. Erd\{o}s also showed that$a_n=\frac{1-c}{n}$with$c>0$does not have this property. Solved by Shepp [Sh72], who showed that a necessary and sufficient condition is that $\sum_n \frac{e^{a_1+\cdots+a_n}}{n^2}=\infty.$ OPEN Let$a_n\in \mathbb{R}$be such that$\sum_n \lvert a_n\rvert^2=\infty$and$\lvert a_n\rvert=o(1/\sqrt{n})$. Is it true that, for almost all$\epsilon_n=\pm 1$, there exists some$z$with$\lvert z\rvert=1$(depending on the choice of signs) such that $\sum_n \epsilon_n a_n z^n$ converges? It is unclear to me whether Erdős also intended to assume that$\lvert a_{n+1}\rvert\leq \lvert a_n\rvert$. It is 'well known' that, for almost all$\epsilon_n=\pm 1$, the series diverges for almost all$\lvert z\rvert=1$(assuming only$\sum \lvert a_n\rvert^2=\infty$). Dvoretzky and Erdős [DE59] showed that if$\lvert a_n\rvert >c/\sqrt{n}$then, for almost all$\epsilon_n=\pm 1$, the series diverges for all$\lvert z\rvert=1$. OPEN Let$f(n,k)$count the number of self-avoiding walks of$n$steps (beginning at the origin) in$\mathbb{Z}^k$(i.e. those walks which do not intersect themselves). Determine $C_k=\lim_{n\to\infty}f(n,k)^{1/n}.$ The constant$C_k$is sometimes known as the connective constant. Hammersley and Morton [HM54] showed that this limit exists, and it is trivial that$k\leq C_k\leq 2k-1$. Kesten [Ke63] proved that$C_k=2k-1-1/2k+O(1/k^2)$, and more precise asymptotics are given by Clisby, Liang, and Slade [CLS07]. Conway and Guttmann [CG93] showed that$C_2\geq 2.62$and Alm [Al93] showed that$C_2\leq 2.696$. OPEN Let$d_k(n)$be the expected distance from the origin after taking$n$random steps from the origin in$\mathbb{Z}^k$(conditional on no self intersections). Is it true that $\lim_{n\to \infty}\frac{d_2(n)}{n^{1/2}}= \infty?$ Is it true that $d_k(n)\ll n^{1/2}$ for$k\geq 3$? OPEN Let$\ell(N)$be maximal such that in any finite set$A\subset \mathbb{R}$of size$N$there exists a Sidon subset$S$of size$\ell(N)$(i.e. the only solutions to$a+b=c+d$in$S$are the trivial ones). Determine the order of$\ell(N)$. In particular, is it true that$\ell(N)\sim N^{1/2}$? Originally asked by Riddell [Ri69]. Erdős noted the bounds $N^{1/3} \ll \ell(N) \leq (1+o(1))N^{1/2}$ (the upper bound following from the case$A=\{1,\ldots,N\}$). The lower bound was improved to$N^{1/2}\ll \ell(N)$by Komlós, Sulyok, and Szemerédi [KSS75]. The correct constant is unknown, but it is likely that the upper bound is true, so that$\ell(N)\sim N^{1/2}$. In [AlEr85] Alon and Erdős make the stronger conjecture that perhaps$A$can always be written as the union of at most$(1+o(1))N^{1/2}$many Sidon sets. (This is easily verified for$A=\{1,\ldots,N\}$using standard constructions of Sidon sets.) OPEN Let$F(k)$be the minimal$N$such that if we two-colour$\{1,\ldots,N\}$there is a set$A$of size$k$such that all subset sums$\sum_{a\in S}a$(for$\emptyset\neq S\subseteq A$) are monochromatic. Estimate$F(k)$. The existence of$F(k)$was established by Sanders and Folkman, and it also follows from Rado's theorem. It is commonly known as Folkman's theorem. Erdős and Spencer [ErSp89] proved that $F(k) \geq 2^{ck^2/\log k}$ for some constant$c>0$. Balogh, Eberhrad, Narayanan, Treglown, and Wagner [BENTW17] have improved this to $F(k) \geq 2^{2^{k-1}/k}.$ SOLVED If$\mathbb{N}$is 2-coloured then is there some infinite set$A\subseteq \mathbb{N}$such that all finite subset sums $\sum_{n\in S}n$ (as$S$ranges over all non-empty finite subsets of$A$) are monochromatic? In other words, must some colour class be an IP set. Asked by Graham and Rothschild. See also [531]. Proved by Hindman [Hi74] (for any number of colours). OPEN Let$\epsilon>0$and$n$be large. Let$G$be a graph on$n$vertices containing no$K_5$and such that every set of$\epsilon n$vertices contains a triangle. Must$G$have$o(n^2)$many edges? The best known result is that$G$can have at most$(\tfrac{1}{12}+o(1))n^2$many edges. OPEN What is the largest possible subset$A\subseteq\{1,\ldots,N\}$which contains$N$such that$(a,b)=1$for all$a\neq b\in A$? A problem of Erdős and Graham. They conjecture that this maximum is either$N/p$(where$p$is the smallest prime factor of$N$) or it is the number of integers$\{2t: t\leq N/2\textrm{ and }(t,N)=1\}$. OPEN Let$r\geq 3$, and let$f_r(N)$denote the size of the largest subset of$\{1,\ldots,N\}$such that no subset of size$r$has the same pairwise greatest common divisor between all elements. Estimate$f_r(N)$. Erdős [Er64] proved that $f_r(N) \leq N^{\frac{3}{4}+o(1)},$ and Abbott and Hanson [AbHa70] improved this exponent to$1/2$. Erdős [Er64] proved the lower bound $f_3(N) > N^{\frac{c}{\log\log N}}$ for some constant$c>0$, and conjectured this should also be an upper bound. Erdős writes this is 'intimately connected' with the sunflower problem [20]. Indeed, the conjectured upper bound would follow from the following stronger version of the sunflower problem: estimate the size of the largest set of integers$A$such that$\omega(n)=k$for all$n\in A$and there does not exist$a_1,\ldots,a_r\in A$and an integer$d$such that$(a_i,a_j)=d$for all$i\neq j$and$(a_i/d,d)=1$for all$i$. The conjectured upper bound for$f_r(N)$would follow if the size of such an$A$must be at most$c_r^k$. The original sunflower proof of Erdős and Rado gives the upper bound$c_r^kk!$. OPEN Let$\epsilon>0$and$N$be sufficiently large. Is it true that if$A\subseteq \{1,\ldots,N\}$has size at least$\epsilon N$then there must be$a,b,c\in A$such that $[a,b]=[b,c]=[a,c],$ where$[a,b]$denotes the least common multiple? This is false if we ask for four elements with the same pairwise least common multiple, as shown by Erdős [Er62]. SOLVED Let$\epsilon>0$and$N$be sufficiently large. If$A\subseteq \{1,\ldots,N\}$has$\lvert A\rvert \geq \epsilon N$then must there exist$a_1,a_2,a_3\in A$and distinct primes$p_1,p_2,p_3$such that $a_1p_1=a_2p_2=a_3p_3?$ A positive answer would imply [536]. Erdős describes a construction of Ruzsa which disproves this: consider the set of all squarefree numbers of the shape$p_1\cdots p_r$where$p_{i+1}>2p_i$for$1\leq i<r$. This set has positive density, and hence if$A$is its intersection with$(N/2,N)$then$\lvert A\rvert \gg N$for all large$N$. Suppose now that$p_1a_1=p_2a_2=p_3a_3$where$a_i\in A$and$p_1,p_2,p_3$are distinct primes. Without loss of generality we may assume that$a_2>a_3$and hence$p_2<p_3$, and so since$p_2p_3\mid a_1\in A$we must have$2<p_3/p_2$. On the other hand$p_3/p_2=a_2/a_3\in (1,2)$, a contradiction. Additional thanks to: Zach Hunter OPEN Let$r\geq 2$and suppose that$A\subseteq\{1,\ldots,N\}$is such that, for any$m$, there are at most$r$solutions to$m=pa$where$p$is prime and$a\in A$. Give the best possible upper bound for $\sum_{n\in A}\frac{1}{n}.$ Erdős observed that $\sum_{n\in A}\frac{1}{n}\sum_{p\leq N}\frac{1}{p}\leq r\sum_{m\leq N^2}\frac{1}{m}\ll r\log N,$ and hence $\sum_{n\in A}\frac{1}{n} \ll r\frac{\log N}{\log\log N}.$ See also [536] and [537]. OPEN Let$h(n)$be such that, for any set$A\subseteq \mathbb{N}$of size$n$, the set $\left\{ \frac{a}{(a,b)}: a,b\in A\right\}$ has size at least$h(n)$. Estimate$h(n)$. Erdős and Szemerédi proved that $n^{1/2} \ll h(n) \ll n^{1-c}$ for some constant$c>0$. SOLVED Is it true that if$A\subseteq \mathbb{Z}/N\mathbb{Z}$has size$\gg N^{1/2}$then there exists some non-empty$S\subseteq A$such that$\sum_{n\in S}n\equiv 0\pmod{N}$? A conjecture of Erdős and Heilbronn. The answer is yes, proved by Szemerédi [Sz70] (in fact for arbitrary finite abelian groups). Erdős speculated that perhaps the correct threshold is$(2N)^{1/2}$; this is also a conjecture of Selfridge, and has been proved when$N$is prime by Balandraud [Ba12]. OPEN Let$a_1,\ldots,a_p$be (not necessarily distinct) residues modulo$p$, such that there exists some$r$so that if$S\subseteq [p]$is non-empty and $\sum_{i\in S}a_i\equiv 0\pmod{p}$ then$\lvert S\rvert=r$. Must there be at most two distinct residues amongst the$a_i$? A question of Graham. SOLVED Is it true that if$A\subseteq\{1,\ldots,n\}$is a set such that$[a,b]>n$for all$a\neq b$, where$[a,b]$is the least common multiple, then $\sum_{a\in A}\frac{1}{a}\leq \frac{31}{30}.$ Is it true that there must be$\gg n$many$m\leq n$which do not divide any$a\in A$? The first bound is best possible as$A=\{2,3,5\}$demonstrates. Resolved by Schinzel and Szekeres [ScSz59] who proved the answer to the first question is yes and the answer to the second is no, and in fact there are examples with at most$n/(\log n)^c$many such$m$, for some constant$c>0$. In [Er73] Erdős further speculates that in fact $\sum_{a\in A}\frac{1}{a}\leq 1+o(1),$ where the$o(1)$term$\to 0$as$n\to \infty$. See also [784]. OPEN Define$f(N)$be the minimal$k$such that the following holds: if$G$is an abelian group of size$N$and$A\subseteq G$is a random set of size$k$then, with probability$\geq 1/2$, all elements of$G$can be written as$\sum_{x\in S}x$for some$S\subseteq A$. Is $f(N) \leq \log_2 N+o(\log\log N)?$ Erdős and Rényi [ErRe65] proved that $f(N) \leq \log_2N+O(\log\log N).$ Erdős believed improving this to$o(\log\log N)$is impossible. OPEN Show that $R(3,k+1)-R(3,k)\to\infty$ as$k\to \infty$. Similarly, prove or disprove that $R(3,k+1)-R(3,k)=o(k).$ This problem is #8 in Ramsey Theory in the graphs problem collection. See also [165]. OPEN Show that if$G$has$\binom{n}{2}$edges then $R(G) \leq R(n).$ More generally, if$G$has$\binom{n}{2}+t$edges with$t\leq n$then $R(G)\leq R(H)$ where$H$is the graph formed by connected a new vertex to$t$of the vertices of$K_n$. In other words, are cliques extremal for Ramsey numbers. Asked by Erdős and Graham. This problem is #10 in Ramsey Theory in the graphs problem collection. SOLVED Is it true that if$G$has$m$edges then $R(G) \leq 2^{O(m^{1/2})}?$ This is true, and was proved by Sudakov [Su11]. The analogous question for$\geq 3$colours is still open. This problem is #11 in Ramsey Theory in the graphs problem collection. Additional thanks to: Zach Hunter OPEN If$T$is a tree on$n$vertices then $R(T) \leq 2n-2.$ Equality holds when$T$is a star on$n$vertices. Implied by [548]. This problem is #14 in Ramsey Theory in the graphs problem collection. OPEN Let$n\geq k+1$. Every graph on$n$vertices with at least$\frac{k-1}{2}n+1$edges contains every tree on$k+1$vertices. A problem of Erdős and Sós, who also conjectured that every graph with at least $\max\left( \binom{2k-1}{2}+1, (k-1)n-(k-1)^2+\binom{k-1}{2}+1\right)$ many edges contains every forest with$k$edges. (Erdős and Gallai [ErGa59] proved that this is the threshold which guarantees containing$k$independent edges.) It can be easily proved by induction that every graph on$n$vertices with at least$n(k-1)+1$edges contains every tree on$k+1$vertices. Brandt and Dobson [BrDo96] have proved this for graphs of girth at least$5$. Wang, Li, and Liu [WLL00] have proved this for graphs whose complements have girth at least$5$. Saclé and Woznik [SaWo97] have proved this for graphs which contain no cycles of length$4$. Yi and Li [YiLi04] have proved this for graphs whose complements contain no cycles of length$4$. Implies [547] and [557]. SOLVED If$T$is a tree which is a bipartite graph with$k$vertices and$2k$vertices in the other class then show that$R(T)=4k$. It follows from results in [EFRS82] that$R(T)\geq 4k-1$. This is false: Norin, Sun, and Zhao [NSZ16] have proved that if$T$is the union of two stars on$k$and$2k$vertices, with an edge joining the centre of the two stars, then$R(T)\geq (4.2-o(1))k$. The best upper bound for the Ramsey number for this tree is$R(T)\leq 4.27492k+1$, obtained by Dubó and Stein [DuSt24]. This problem is #15 in Ramsey Theory in the graphs problem collection. Additional thanks to: Zach Hunter OPEN Let$m_1\leq\cdots\leq m_k$and$n$be sufficiently large. If$T$is a tree on$n$vertices and$G$is the complete multipartite graph with vertex class sizes$m_1,\ldots,m_k$then prove that $R(T,G)\leq (\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1.$ Chvátal [Ch77] proved that$R(T,K_m)=(m-1)(n-1)+1$. This problem is #16 in Ramsey Theory in the graphs problem collection. SOLVED Prove that $R(C_k,K_n)=(k-1)(n-1)+1$ for$k\geq n\geq 3$(except when$n=k=3$). Asked by Erdős, Faudree, Rousseau, and Schelp, who also ask for the smallest value of$k$such that this identity holds (for fixed$n$). They also ask, for fixed$n$, what is the minimum value of$R(C_k,K_n)$? This identity was proved for$k>n^2-2$by Bondy and Erdős [BoEr73]. Nikiforov [Ni05] extended this to$k\geq 4n+2$. Keevash, Long, and Skokan [KLS21] have proved this identity when$k\geq C\frac{\log n}{\log\log n}$for some constant$C$, thus establishing the conjecture for sufficiently large$n$. OPEN Determine $R(C_4,S_n),$ where$S_n$is the star on$n+1$vertices. It was shown in [BEFRS89] that $n+\lceil\sqrt{n}\rceil+1\geq R(C_4,S_n)\geq n+\sqrt{n}-6n^{11/40}.$ Füredi (unpublished) has shown that$R(C_4,S_n)=n+\lceil\sqrt{n}\rceil$for infinitely many$n$. SOLVED Let$R(3,3,n)$denote the smallest integer$m$such that if we$3$-colour the edges of$K_m$then there is either a monochromatic triangle in one of the first two colours or a monochromatic$K_n$in the third colour. Define$R(3,n)$similarly but with two colours. Show that $\frac{R(3,3,n)}{R(3,n)}\to \infty$ as$n\to \infty$. A problem of Erdős and Sós. This was solved by Alon and Rödl [AlRo05], who in fact show that $R(3,3,n)\asymp n^3(\log n)^{O(1)}$ (recalling that Shearer [Sh83] showed$R(3,n) \ll n^2/\log n$). OPEN Let$R(G;k)$denote the minimal$m$such that if the edges of$K_m$are$k$-coloured then there is a monochromatic copy of$G$. Show that $\lim_{k\to \infty}\frac{R(C_{2n+1};k)}{R(K_3;k)}=0$ for any$n\geq 2$. A problem of Erdős and Graham. The problem is open even for$n=2$. OPEN Let$R(G;k)$denote the minimal$m$such that if the edges of$K_m$are$k$-coloured then there is a monochromatic copy of$G$. Determine the value of $R(C_{2n};k).$ A problem of Erdős and Graham. Erdős [Er81c] gives the bounds $k^{1+\frac{1}{2n}}\ll R(C_{2n};k)\ll k^{1+\frac{1}{n-1}}.$ Chung and Graham [ChGr75] showed that $R(C_4;k)>k^2-k+1$ when$k-1$is a prime power and $R(C_4;k)\leq k^2+k+1$ for all$k$. SOLVED Let$R(G;3)$denote the minimal$m$such that if the edges of$K_m$are$3$-coloured then there must be a monochromatic copy of$G$. Show that $R(C_n;3) \leq 4n-3.$ A problem of Bondy and Erdős. This inequality is best possible for odd$n$. Luczak [Lu99] has shown that$R(C_n;3)\leq (4+o(1))n$for all$n$, and in fact$R(C_n;3)\leq 3n+o(n)$for even$n$. Kohayakawa, Simonovits, and Skokan [KSS05] proved this conjecture when$n$is sufficiently large and odd. Benevides and Skokan [BeSk09] proved that if$n$is sufficiently large and even then$R(C_n;3)=2n$. OPEN Let$R(G;k)$denote the minimal$m$such that if the edges of$K_m$are$k$-coloured then there is a monochromatic copy of$G$. Is it true that $R(T;k)=kn+O(1)$ for any tree$T$on$n$vertices? A problem of Erdős and Graham. Implied by [548]. OPEN Let$R(G;k)$denote the minimal$m$such that if the edges of$K_m$are$k$-coloured then there is a monochromatic copy of$G$. Determine $R(K_{s,t};k)$ where$K_{s,t}$is the complete bipartite graph with$s$vertices in one component and$t$in the other. Chung and Graham [ChGr75] prove the bounds $(2\pi\sqrt{st})^{\frac{1}{s+t}}\left(\frac{s+t}{e^2}\right)k^{\frac{st-1}{s+t}}\leq R(K_{s,t};k)\leq (t-1)(k+k^{1/s})^s.$ For example this implies that $R(K_{3,3};k) \ll k^3.$ Using Turán numbers one can show that $R(K_{3,3};k) \gg \frac{k^3}{(\log k)^3}.$ SOLVED Let$\hat{R}(G)$denote the size Ramsey number, the minimal number of edges$m$such that there is a graph$H$with$m$edges that is Ramsey for$G$. If$G$has$n$vertices and maximum degree$d$then prove that $\hat{R}(G)\ll_d n.$ A problem of Beck and Erdős. Beck [Be83b] proved this when$G$is a path. Friedman and Pippenger [FrPi87] proved this when$G$is a tree. Haxell, Kohayakawa, and Luczak [HKL95] proved this when$G$is a cycle. An alternative proof when$G$is a cycle (with better constants) was given by Javadi, Khoeini, Omidi, and Pokrovskiy [JKOP19]. This was disproved for$d=3$by Rödl and Szemerédi [RoSz00], who constructed a graph on$n$vertices with maximum degree$3$such that $\hat{R}(G)\gg n(\log n)^{c}$ for some absolute constant$c>0$. Tikhomirov [Ti22b] has improved this to $\hat{R}(G)\gg n\exp(c\sqrt{\log n}).$ It is an interesting question how large$\hat{R}(G)$can be if$G$has maximum degree$3$. Kohayakawa, Rödl, Schacht, and Szemerédi [KRSS11] proved an upper bound of$\leq n^{5/3+o(1)}$and Conlon, Nenadov, and Trujić [CNT22] proved$\ll n^{8/5}$. The best known upper bound of$\leq n^{3/2+o(1)}$is due to Draganić and Petrova [DrPe22]. Additional thanks to: Zach Hunter OPEN Let$\hat{R}(G)$denote the size Ramsey number, the minimal number of edges$m$such that there is a graph$H$with$m$edges such that in any$2$-colouring of the edges of$H$there is a monochromatic copy of$G$. Determine $\hat{R}(K_{n,n}),$ where$K_{n,n}$is the complete bipartite graph with$n$vertices in each component. We know that $\frac{1}{60}n^22^n<\hat{R}(K_{n,n})< \frac{3}{2}n^32^n.$ The lower bound (which holds for$n\geq 6$) was proved by Erdős and Rousseau [ErRo93]. The upper bound was proved by Erdős, Faudree, Rousseau, and Schelp [EFRS78b] and Nešetřil and Rödl [NeRo78]. Conlon, Fox, and Wigderson [CFW23] have proved that, for any$s\leq t$, $\hat{R}(K_{s,t})\gg s^{2-\frac{s}{t}}t2^s,$ and prove that when$t\gg s\log s$we have$\hat{R}(K_{s,t})\asymp s^2t2^s$. They conjecture that this should hold for all$s\leq t$, and so in particular we should have$\hat{R}(K_{n,n})\asymp n^32^n$. OPEN Let$\hat{R}(G)$denote the size Ramsey number, the minimal number of edges$m$such that there is a graph$H$with$m$edges such that in any$2$-colouring of the edges of$H$there is a monochromatic copy of$G$. Let$F_1$and$F_2$be the union of stars. More precisely, let$F_1=\cup_{i\leq s} K_{1,n_i}$and$F_2=\cup_{j\leq t} K_{1,m_j}\$. Prove that $\hat{R}(F_1,F_2) = \sum_{2\leq k\leq s+2}\max\{n_i+m_j-1 : i+j=k\}.$