124 solved out of 520 shown

$500

If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in A'}a$ are distinct for all $A'\subseteq A$ 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 $\{1,\ldots,Nn\}$. 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 $\sqrt{2/\pi}$ due to Dubroff, Fox, and Xu [DFX21].

In [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.

$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] have proved $r_4(N)\ll N/(\log N)^{c}$ for some small constant $c>0$. The best bound available for general $k$ is due to Gowers [Go01], \[r_k(N) \ll \frac{N}{(\log\log N)^{c_k}},\] where $c_k>0$ is a small constant 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].

See also [142].

$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], which showed the existence of some constant $C>0$ such that the claim holds. Solved by Maynard [Ma16] and Ford, Green, Konyagin, and Tao [FoGrKoTa16]. The best bound available, due to all five authors [FoGrKoMaTa18], 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 $c>0$.

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.

$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 \[d_n> d_{n+1}>\cdots >d_{n+m}.\]

Is there a covering system all of whose moduli are odd?

Asked by Erdős and Selfridge (sometimes also with Schinzel). They also ask can there be such that no two of the moduli divide each other, or where all the moduli are odd and square-free?

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 $n_j$.

Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST21] have proved that if the moduli are all squarefree then at least one must be even.

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.

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 $\{1,\ldots,N\}$. 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.

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 $1-\epsilon$.

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 $p$).

Erdős thinks that proving this with two powers of 2 is perhaps easy.

Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$. Is there such an $A$ with
\[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>0?\]

Asked by Erdős and Sárközy, who proved that $A$ must have density $0$. The set of $p^2$, where $p\equiv 3\pmod{4}$ is prime, has this property. Note that such a set cannot contain a three-term arithmetic progression, and hence by the bound of Kelley and Meka [KeMe23] we have
\[\lvert A\cap\{1,\ldots,N\}\rvert\ll \exp(-O((\log N)^{1/12}))N\]
for all large $N$.

$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].

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 a set which shows this would be the best possible.

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.

$500

If $G$ is an edge-disjoint union of $n$ copies of $K_n$ then is $\chi(G)=n$?

$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, 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 $C>1$. 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
\[f(n,k)=(1+O_n(k^{-1/2^n}))k^n.\]

Is it true that for every $C>0$ if $n$ is large enough and $\mathcal{F}$ is an intersecting family of sets of size $n$ with $\lvert \mathcal{F}\rvert \leq Cn$ then there exists a set $S$ with $\lvert S\rvert \leq n-1$ which intersects every $A\in\mathcal{F}$?

Conjectured by Erdős and Lovász [ErLo75], who proved that this holds if $\lvert \mathcal{F}\rvert\leq \frac{8}{3}n-4$. Disproved by Kahn [Ka94] who constructed an infinite sequence of $\mathcal{F}$, each a family of sets of size $n\to\infty$, such that any set $S$ of size $n-1$ is disjoint from at least one set in $\mathcal{F}$. The Erdős and Lovász constant of $8/3$ has not been improved.

Is there, for every $\epsilon>0$, 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 $\epsilon n$?

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
\[ \ll \frac{(\log\log n)^{3/2}}{(\log n)^{1/2}}n.\]

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.

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].

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 has found a counterexample (according to Erdős in [Er95], although I cannot find a paper in which this appears). 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 $a\in A$.

$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 $N\leq n_1$ and $n_k\leq Cn_1$?

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 $n_1<10^{18}$.
[NOTE: This is my best attempt at recovering problem 5 from [Er95], which doesn't make sense as written.]

$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 $\lvert A\cap [1,N]\rvert \gg N^{1/2}$ for all large $N$ implies $\limsup 1_A\ast 1_A(n)=\infty$. 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 $\limsup 1_A\ast 1_B(n)=\infty$.

$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.

$1000

Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$,
\[h(N) = N^{1/2}+O_\epsilon(N^\epsilon)?\]

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]. 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 $N$.

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 $a\in A$?

Erdős [Er54] proved by the probabilistic method that such a set $A$ exists with $\lvert A\cap\{1,\ldots,N\}\rvert\ll (\log N)^2$.

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 $>1$.

Let $A\subseteq\mathbb{N}$ be an additive basis of order $k$. Is it true that for every $B\subseteq\mathbb{N}$ 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 proved this is true with $k$ replaced by $2k$ in the denominator.

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 $\| 1_A\circ 1_B\|_\infty \geq cN$ - that is, 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].

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 $A\subset\mathbb{N}$ (i.e. there exists some $\lambda>1$ such that $A=\{n_1<n_2<\cdots\}$ satisfies $n_{k+1}>\lambda n_k$) 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 $N$.

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 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 $1_A\ast 1_A(n)=O(1)$.

$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
\[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0.\]

$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 $c>0$?

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]. Obtaining better quantitative aspects seems interesting still -- note that Croot's result allows for $n_k \leq e^{C^k}$ for some constant $C>1$. Is there a better bound?

$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. Erdős further speculates 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.)

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 \[ \lvert A\rvert \leq \left(1+O\left(\frac{(\log\log x)^5}{\log x}\right)\right)\pi(x).\]

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 $t$.

$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 $1558/1167=1.33504\cdots$).

$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 $A$. It is known that it cannot be true that
\[\lvert A\cap \{1,\ldots,N\}\rvert \ll (\log N)^2\]
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].

$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.

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 $k\geq 8$ by Khachatrian [AhKh94]. Erdős asks if the conjecture remains true provided $N\geq (1+o(1))p_k^2$.

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 Hajnal and Erdős and solved by Liu and Montgomery [LiMo20]. The lower density of the set can be $0$ since there are graphs of arbitrarily large chromatic number and girth.

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.

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 $G$.

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 $q$.

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.

$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 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 $m$.

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 the sum $\sum\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph?

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 $\geq (\tfrac{1}{2}-o(1))\log k$.

$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.

$500

If $f:\mathbb{N}\to \{-1,+1\}$ then is it true that for every $C>0$ there exist some $d,m\geq 1$ 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.

Is it true that for every pair $a,b\geq 1$ such that either $a$ is even or both $a$ and $b$ are odd then there is $c=c(a,b)$ such that every graph with average degree at least $c$ contains a cycle whose length is $\equiv a\pmod{b}$?

This has been proved by Bollobás [Bo77]. The best dependence of the constant $c(a,b)$ is unknown.

$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 proved that such a $c$ does exist if $A$ is an infinite arithmetic progression containing even numbers. Erdős was 'almost certain' that if $A$ is the set of powers of $2$ then no such $c$ exists (although conjectures 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.

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.)

$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 [ErHaSz82]. Rödl [Ro82] has proved this for hypergraphs. 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 $\aleph_1$.

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 [ErHaSz82].

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].

Erdős offers \$100 for just a proof of the existence of this constant, without determining its value. He also offers \$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 [CaGrMoSa23].

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 $C>0$.

We say $G$ is Ramsey size linear if $R(G,H)\ll e(H)$ for all graphs $H$ with 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 [ErFaRoSc93]. $K_4$ is the only known example of such a graph.

In other words, is $K_{3,3}$ Ramsey size linear? Asked by Erdős, Faudree, Rousseau, and Schelp [ErFaRoSc93].

Let $G$ be a graph on $n$ vertices with 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 [ErOrZa93], who proved an upper bound of $(1/4-\epsilon)n^2$ many cliques (for some very small $\epsilon>0$).

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 Staton. 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 $n^{1/2}(\log n)^{O(1)}$.

$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 $\{1,\ldots,t+2r\}$.

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 $f(n)=o(2^n)$.

Conjectured by Erdős and Faudree, who showed that $f(n) > 2^{n/2}$, and further speculate that $f(n)/2^{n/2}\to \infty$. This conjecture was proved by Verstraëte [Ve04], who proved the number of such sets is
\[\ll 2^{n-n^c}\]
for some constant $c>0$.

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 $C_4$.

$100

Is it true that every subgraph of the $n$-dimensional cube 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.

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 $k=4$.

Since $R(k)\leq 4^k$ this is trivial for $\epsilon\geq 1/4$. 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 $R(k)$.

$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.

$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 possibe. Nearly solved by Guth and Katz [GuKa15] who proved that there are always $\gg n/\log n$ many distinct distances. It may be true that there is a single point which determines $\gg n/\sqrt{\log n}$ distinct distances, or even that there are $\gg n$ many such points, or even that this is true averaged over all points.

$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?

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 [SpSzTr84]. In [Er83c] Erdős offers \$250 for an upper bound of the form $n^{1+o(1)}$.

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').

If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then they determine at least $\lfloor n/2\rfloor$ distinct distances.

Solved by Altman [Al63]. The stronger variant that says there is one point which determines at least $\lfloor n/2\rfloor$ distinct distances is still open.

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.

$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 $\epsilon>0$
\[\sum_i f(u_i)^2 \ll_\epsilon n^{3+\epsilon}.\]

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 has proved an upper bound of $O(n\log n)$. Edelsbrunner and Hajnal have constructed $n$ such points with $2n-7$ pairs distance $1$ apart.

Does every polygon have a vertex with no other $4$ vertices equidistant from it?

Open even for convex polygons.

Danzer has 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 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?

$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?

In fact Erdős believes such a set must have very large intersection with the triangular lattice. This is false for $n=4$, for example a square. The behaviour of such sets for small $n$ is explored by Bezdek and Fodor [BeFo99].

Let $A\subseteq\mathbb{R}^2$ be a set of $n$ points with distance set $D=\{\lvert x-y\rvert : x\neq y \in A\}$. Suppose that if $d_1\neq d_2\in D$ then $\lvert d_1-d_2\rvert \geq 1$. Is there some constant $c>0$ such that the diameter of $A$ must be at least $cn$?

Perhaps the diameter is $\geq n-1$ for all large $n$ (Piepmeyer has an example of $9$ such points with diameter $<5$).

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 $n$. 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 $\gg n/\log n$.

$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 $B$?

$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
\[f(n) \leq 2^{n+O(\sqrt{n\log n})}.\]

For every $k\geq 3$ and $n\geq 2$ is there some finite $f(n,k)$ such that every graph of chromatic number $\geq f(n,k)$ contains a subgraph of girth at least $k$ and chromatic number at least $n$?

Conjectured by Erdős and Hajnal. Rödl [Ro77] has proved the $k=3$ 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.

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].

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 [ErHaSz82]. This fails if the graph has chromatic number $\aleph_0$.

Let $c>0$ and $G$ be a graph of chromatic number $\aleph_1$. Are there infinitely many $n$ such that $G$ contains a subgraph on $n$ vertices which cannot be made bipartite by deleting at most $cn$ edges?

Conjectured by Erdős, Hajnal, and Szemerédi [ErHaSz82].

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 $p(z)=z^n-1$?

If $p(z)$ is a polynomial of degree $n$ then
\[\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2.\]
What if the set $\{z : \lvert p(z)\rvert\leq 1\}$ is connected?

The lower bound is easy: this is $\geq n$ and equality holds if and only if $p(z)=z^n$. 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] has proved an upper bound of $O(n^2)$, and $\frac{e}{2}n^2$ if the set is connected.

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 $\epsilon>0$.

Let $G$ be a group and let $\Gamma(G)$ be the commuting graph of $G$, with vertex set $G$ and $x\sim y$ if and only if $xy=yx$. Let $h(n)$ be minimal such that if the largest independent set of $\Gamma(G)$ has at most $n$ elements then $\Gamma(G)$ can be covered by $h(n)$ many complete subgraphs. Estimate $h(n)$ as well as possible.

Pyber [Py87] has proved there exist constants $c_2>c_1>1$ such that $c_1^n<h(n)<c_2^n$.

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.

$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 \[\max_{n\leq N} M_n > N^c.\]

$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.

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 [ErSaSo95], who proved this for $F_3(N)$. Erdős [Er35] earlier proved that $F_4(N)=o(N)$. Erdős also asks about $F(N)$, the size of the largest such set such that the product of no odd number of $a\in A$ is a square. Ruzsa proved that $\lim F(N)/N <1$. The value of $\lim F(N)/N$ is unknown, but it is $>1/2$.

Let $f(n)$ be a number theoretic function which grows slowly 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] prove this when $f$ is the divisor function or the number of distinct prime divisors of $n$, but Erdős believes it is false when $f(n)=\phi(n)$ or $\sigma(n)$.

$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 $c=7$.

Let $d_1<d_2<\cdots <d_k$ 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 form $\sum_{1\leq i\leq k}a_i$, where $a_i$ has only the digits $0,1$ when written in base $d_i$?

Conjectured by Burr, Erdős, Graham, and Li [BuErGrLi96]. Pomerance observed that the condition $\sum 1/(d_i-1)\geq 1$ is necessary. In [BuErGrLi96] they prove the property holds for $\{3,4,7\}$.

Let $A\subseteq \mathbb{N}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B\subseteq \mathbb{N}$ be the set of integers which have only the digits $0,1$ when written base $4$. Does $A+B$ have positive density?

Conjectured by Burr, Erdős, Graham, and Li [BuErGrLi96].

$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] 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.

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.

$250

Let $G$ be a graph with $10n$ vertices such that every subgraph on $5n$ vertices has more than $2n^2$ many edges. Must $G$ contain a triangle?

A blown-up $C_5$ (or, as Simonovits observed, a blown-up Petersen graph) shows that this would be best possible.

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. It is 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: 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), as soon as $N \geq C^n$ 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.

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?

Conjectured by Andrásfai and Erdős. It is possible that such a graph could contain an infinite complete graph.

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 $\lvert A\rvert \gg N^{1/5}$.

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?

Asked by Erdős and Pach. Pannowitz proved that this is true for the largest distance between points of $A$, but it is unknown whether a second such distance must occur.

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. 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 $\binom{2m-1}{m}$, showing that $f(n)=o(n)$ infinitely often. This graph may have the minimal possible $f$, but Erdős encourages the reader to try and find a better graph.

Let $\epsilon,\delta>0$ and $n$ 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 $C$.

$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.

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.

$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 $k=3$, Green and Tao [GrTa17] for $k=4$, Leng, Sah, and Sawhney [LSS23] for $k=5$, and Gowers [Go01] for $k\geq 6$.

$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 $C>0$.

$1000

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 $r_k(N)$.

The best known bounds are due to Kelley and Meka [KeMe23] for $k=3$, Green and Tao [GrTa17] for $k=4$, and Gowers [Go01] for $k\geq 5$. Erdős remarked this is 'probably unattackable at present'. Given that Erdős elsewhere offered \$5000 just for (essentially) showing that $r_k(N)=o_k(N/\log N)$, the value of this prize seems odd.

See also [3].

$500

Let $A\subset \mathbb{R}$ be a countably infinite set (with $1\not\in A$) 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
\[\sum_{x\in A}\frac{1}{x\log x}<\infty,\]
or
\[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n)?\]

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].

For every $\epsilon>0$ there is a polynomial $P(z)=\prod_{i=1}^n(z-\alpha_i)$ with $\lvert \alpha_i\rvert=1$ such that the measure of the set for which $\lvert P(z)\rvert<1$ is at most $\epsilon$.

Erdős says it would also be of interest to determine the dependence of $n$ on $\epsilon$ - perhaps $\epsilon>1/(\log n)^c$ for some $c>0$ is necessary.

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.

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.

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 clique 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'.

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.

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.

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 [ErFaRoSc78]), and the lower bound is due to Spencer [Sp77].

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 $c>0$.

$500

Let $\alpha>0$ and $n,t\geq 1$ be integers. Let $F^{(t)}(n,\alpha)$ be the largest $k$ such that the following holds.

Let $\lvert S\rvert=n$ and 2-colour all $t$-subsets of $S$. For every $X\subseteq S$ of size at least $k$ 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, Rado, and Hajnal 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 $1/2$.

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
\[c_1(\alpha)\log n < F(n,\alpha) < c_2(\alpha)\log n.\]

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. Solved by Lee [Le16], who proved that
\[ R(H) \leq 2^{2^{O(d)}}n.\]

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.

Spencer [Sp77] proved
\[R(4,k) \gg (k\log k)^{5/2}.\]
Ajtai, Komlós, and Szemerédi [AjKoSz80] 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
\[R(4,k) \gg \frac{k^3}{(\log k)^4}.\]

Estimate the hypergraph Ramsey number $R_r(k)$. Does it grow like
\[2^{2^{\cdots k}}\]
where the tower of exponentials has height $r-1$?

Erdős, Hajnal, and Rado proved that $R_r(k)$ grows like a tower of exponentials of height at least $r-2$ and at most $r-1$. Hajnal has proved that $r-1$ is the correct height if we consider the corresponding hypergraph Ramsey number for $4$ colours.

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.

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.

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 $A\subset \{0,1,\ldots,N\}$.

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 $\lvert A\rvert=2$.

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.

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].

Show that, for any $n\geq 5$, the binomial coefficient $\binom{2n}{n}$ is not squarefree.

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 $f(n)\gg \log n$?

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 $C_c>1$.

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 $h(d)\gg d^{1/2}$.

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.

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 some constant $C_{\mathcal{F}}>0$ and $G\in\mathcal{F}$ such that for all large $n$ \[\mathrm{ex}(n;G)\leq C_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?\]

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.

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.)

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?

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.

$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],
\[R(3;k) \gg (321)^{k/5}.\]

See also [483].

Any graph on $n$ vertices can be decomposed into $O(n)$ many cycles and edges.

Conjectured by Erdős and Gallai. 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.

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].

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 $\ll (N\log N)^{1/2}$).

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 $f(d) \ll d$. 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.

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').

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.

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 $H(k)^{1/k}\to\infty$.

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 \[\sum_{x\in X}\frac{1}{\log x}\geq 2^{-8}\log\log\log n.\]

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?

It is known that it must contain a 4-term arithmetic progression if $d=2$, and need not contain a 3-term arithmetic progression if $d=5$. The cases $d=3$ and $d=4$ are unknown.

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 $\mathbb{Z}^3$, and for $\mathbb{Z}^2$ that the number of collinear points can be bounded.

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].

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.

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
\[\lim_{N\to \infty}\frac{R_3(N)}{G_3(N)}=1?\]

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 $R_k(N) \ll_k G_k(N)$.

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.

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 $2^{2^t}+1$.

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.

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.

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].

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. The current best bound is due to Chan [Ch21], who proved an upper bound of $s_n^{\frac{5}{26}+o(1)}$. More precisely, they proved that there exists some $C>0$ such that, for all large $x$, the interval
\[(x,x+Cx^{5/26}]\]
contains a squarefree number.

See also [489].

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': three lines from $A$ which intersect in three points, and each of these intersection points only intersects two lines from $A$?

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 $d\geq 4$.

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 intersect exactly 2 points. Does $f(n)\to \infty$? How fast?

Conjectured by Erdős and de Bruijn. The Sylvester-Gallai theorem states that $f(n)\geq 1$. 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 $n$.

$100

Let $1\leq k<n$. Given $n$ points in $\mathbb{R}^2$, at most $n-k$ on any line, the number of distinct lines formed joining two points is $\gg kn$.

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].

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).

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?

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, hten \[\lvert S\rvert \ll (\log N)^{O(1)}.\]

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.

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.

Let $g(k)$ be the smallest integer (if any such exists) such that any $g(k)$ points in $\mathbb{R}^2$ contains a convex $k$-gon with no point in the interior. Does $g(k)$ exist? If so, estimate $g(k)$.

A variant of the 'happy ending' problem, 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 $n\geq 7$.

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 'a Hungarian high school student' gave an unspecified construction for $n=6$. Is this possible for $n=7$?

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 $\gamma\geq 1$.

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] showed there is such a set with, for all large $N$,
\[\lvert A\cap\{1,\ldots,N\}\rvert \ll \frac{\log\log N}{\log N}N\]

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
\[n_{k+1}-n_k \ll n_k^{1/4}.\]

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].

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 $c_k\in \mathbb{C}$).

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 $f(A)=B$?

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].

See also [230].

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
\[f^{(k_n)}(z) = 0\textrm{ for all }z\in S_n?\]

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 forall $z\in\mathbb{C}$ 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 $\lvert z\rvert=1$.

See also [228].

Let $S$ be a string of length $2^k-1$ formed from an alphabet of $k$ characters. Must there be two consecutive blocks each of which contain the same number of each symbol in the alphabet?

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. I do not know the status of this question for $k\geq 5$. Is it true that there is in fact an infinite string formed from $\{1,2,3,4\}$ which contains no consecutive finite blocks each of which contain the same number of each symbol?

Containing no such consecutive blocks 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).

Let $k\geq 1$ and define $N(k)$ to be the minimal $N$ such that any string $s\in \{1,\ldots,k\}^N$ contains two adjacent blocks such that each is a rearrangement of the other. Estimate $N(k)$.

Erdős originally conjectured that $N(k)=2^k-1$, but this was disproved by Erdős and Bruijn. It is not even known whether $N(4)$ is finite.

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).

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 Halász [Ha68].

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.

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 incidences). 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 $(7/2)^{1/3}\approx 1.519\cdots$).

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 \[\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.\]

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.

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].

Is
\[\sum_n \frac{\phi(n)}{2^n}\]
irrational? Here $\phi$ is the Euler totient function.

Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$. Is
\[\sum \frac{\sigma_k(n)}{n!}\]
irrational?

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].

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 $\theta\in (0,1)$.

Let $n\geq 1$ and $f(n)$ be maximal such that, for every set $A\subset \mathbb{N}$ with $\lvert A\rvert=n$, we have
\[\max_{\lvert z\rvert=1}\left\lvert \prod_{n\in A}(1-z^n)\right\rvert\geq f(n).\]
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 $c>0$.

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
- $a_n \gg n\sqrt{\log n\log\log n}$.

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].

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 larger constant then higher degree polynomials can be used - for example if we consider $\sum\frac{1}{a_n}$ and $\sum\frac{1}{a_n+8}$ then $a_n=n^3+6n^2+5n$ is an example.

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 $\sum_{n\in A}\frac{1}{n}<\infty$.

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.

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.

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.

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.

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].

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'.

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.

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).

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 $\sum \frac{1}{n_i}=\infty$.

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.

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 $1=\sum \frac{1}{n_i}$.

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].

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.

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,
\[\sum_{3\leq n\leq 5}\frac{1}{n} = \frac{47}{60}\textrm{ and }\sum_{3\leq n\leq 6}\frac{1}{n}=\frac{19}{20}.\]

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.

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'.

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] claim a bound of
\[t(N)\ll\frac{N}{\log N},\]
but have no idea of the true value of $t(N)$.

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
\[-c < k(N)-(e-1)N \ll \frac{N}{\log N}.\]

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 $k(N)=(1-o(1))\log N$.

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 $A(N)\geq (1-\frac{1}{e}+o(1))N$.

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 $\lvert A\rvert\geq N/2$.

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.

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].

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
\[D(p) \gg p\log p.\]

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 [BuErGr15] (this paper is perhaps Erdős' last paper, appearing 19 years after his death).

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 $Q$?

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?

Hunter and Sawhney have observed that Theorem 3 of Bloom [Bl23] implies the answer to the second question is no: in fact all integers $m\leq (1-o(1))\log N$ can be written as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$.

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 $c/K^2$.

Let $n\geq 1$ and let $t$ be minimal such that $\sum_{n\leq k\leq t}\frac{1}{k}\geq 1$. We define
\[\epsilon(n) = \sum_{n\leq k\leq t}\frac{1}{k}-1.\]
How small can $\epsilon(n)$ be? Is it true that
\[\liminf n^2\epsilon(n)=0?\]

It is perhaps true that $n^{2+\delta}\epsilon(n)\to \infty$ for all $\delta >0$.

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.

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.

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
\[\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.\]

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].

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].

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 $k$).

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 $f_{k,k}(x)=o(x)$.

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].

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 $a+b\mid ab$.

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 $C$.

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 $D(A)\neq\emptyset$.

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 $f(n)\leq e^{O(\sqrt{\log n})}$.

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)?

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 $\geq 7$).

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.

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?

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.

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.

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.

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.

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].

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 $2^{t+1}$.

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 $B$.

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.

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.

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.

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].

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?

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.

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.

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.

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 latter problem has been solved by Konieczny [Ko15], who showed that a random permutation of $\{1,\ldots,n\}$ has, with high probability, at least
\[\left(\frac{1+e^{-2}}{4}+o(1)\right) n^2\]
many such consecutive sums.

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 $n$)?

Let $1\leq a_1<\cdots <a_k\leq n$ are 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 $\sum_i \frac{1}{a_i}\ll 1$?

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.

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 $\sum_{n\in S}n=t$?

Erdős and Moser proved this bound with an additional factor of $(\log n)^{3/2}$. This was removed by Sárközy and Szemerédi [SaSz65]. Stanley [St80] has shown that this quantity is maximised when $A$ is an arithmetic progression and $t=\tfrac{1}{2}\sum_{n\in A}n$.

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.

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 $x^2=8y^2+1$.

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 $x^2=8y^2+1$.

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?

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 $\epsilon>0$,
\[\limsup \frac{\prod_{n\leq m<n+k}B_r(m) }{n^{1+\epsilon}}\to\infty?\]

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?

Open even for $k=2$. Erdős and Graham remark 'the answer should be affirmative but the problem seems very hard'.

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.

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 $n=a_2!\cdots a_k!$.

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 $k>6$.

Is
\[\sum_{\substack{p\nmid \binom{2n}{n}\\ p\leq n}}\frac{1}{p}\]
unbounded as a function of $n$?

If the sum is denoted by $f(n)$ then Erdős, Graham, Ruzsa, and Straus [EGRS75] have shown
\[\lim_{x\to \infty}\frac{1}{x}\sum_{n\leq x}f(n) = \sum_{k=2}^\infty \frac{\log k}{2^k}=\gamma_0\]
and
\[\lim_{x\to \infty}\frac{1}{x}\sum_{n\leq x}f(n)^2 = \gamma_0^2,\]
so that for almost all integers $f(m)=\gamma_0+o(1)$.

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.

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 $s(n) \asymp \log n$).

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].

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].

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 $c>0$.

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 $3\leq k\leq n-3$.'

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].

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.

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 $a_t$.

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).

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 $k\geq 3$.

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.

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 $(x,y)=1$.

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].

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) then the only cases where equality 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].

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].

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 $m_k\to \infty$?

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 $\leq n^{o(1)}$.

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 $k\geq 6$.

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 $\sigma_i(m)=\sigma_j(n)$?

That is, there is (eventually) only one possible sequence that the iterated sum of divisors function can settle on. Selfridge reports numerical evidence suggests the answer is no, but Erdős and Graham write 'it seems unlikely that anything can be proved about this in the near future'.

Let $\nu(n)$ count the number of distinct primes dividing $n$. Are there infinitely many $n$ such that, for all $m<n$, we have $m+\nu(m) \leq n$?

Erdős suggested this problem as a way of showing that the iterated behaviour of $n\mapsto n+\nu(n)$ 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'. Can one show that there exists an $\epsilon>0$ such that there are infinitely many $n$ where $m+\epsilon \nu(m)\leq n$ for all $m<n$?

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.

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 $V(x)=o(x)$. The behaviour of $V(x)$ is almost completely understood. Maier and Pomerance [MaPo88] proved
\[V(x)=\frac{x}{\log x}e^{(C+o(1))\log\log\log x},\]
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 $V(2x)/V(x)\to 2$.

See also [417].

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 $\sigma(n)-n$.

If $\tau(n)$ counts the number of divisors of $n$, then what is the set of limit points of
\[\frac{\tau((n+1)!)}{\tau(n!)}?\]

It can be shown that any number of the shape $1+1/k$ for $k\geq 1$ is a limit point (and thus so too is $1$), but no others are known.

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 $c>0$.

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 $n$.

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.

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 almost all integers appear in this sequence?

Asked by Hofstadter. The sequence begins $2,3,5,9,14,17,26,\ldots$ and is A005244 in the OEIS.

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
\[F(n)=\pi(n)+(c+o(1))n^{3/4}(\log n)^{-3/2}?\]

Erdős [Er68] proved that
\[F(n)-\pi(n)\asymp n^{3/4}(\log n)^{-3/2}.\]
Surprisingly, if we consider the corresponding problem in the reals (so the largest $m$ such that there are reals $1\leq a_1<\cdots <a_m\leq x$ such that for any distinct indices $i,j,r,s$ we have $\lvert a_ia_j-a_ra_s\rvert \geq 1$) then Alexander proved that $m> x/8e$ is possible.

See also [490].

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 $\limsup$.

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.

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.

See also [432].

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 $t\to \infty$.

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 $\lvert A\rvert\leq (\tfrac{11}{32}+o(1))N$.

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].

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].

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 $\delta'(n)=o(\delta(n))$?

Erdős [Er35] proved that $\delta(n)<(\log n)^{-c}$ for some $c>0$. Tenenbaum [Te75] showed that $\delta(n)=(\log n)^{(-1+o(1))\alpha}$ where
\[\alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\cdots.\]

This has been resolved by Ford [Fo08]. Among many other results, Ford proves \[\delta(n)\asymp \frac{1}{(\log n)^\alpha(\log\log n)^{3/2}},\] and that the second conjecture is false.

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 \[\lvert \mathcal{F}\rvert <(1+o(1))\binom{n}{\lfloor n/2\rfloor}.\]

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 $n$?

Erdős and Graham also ask whether there is a good inequality known for $\sum_{n\leq x}\tau^*(n)$.

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 $k$.

Let $A_n$ be the least common multiple of $\{1,\ldots,n\}$. Is it true that, for all $k\geq 1$,
\[A_{p_{k+1}-1}< p_kA_{p_k}?\]

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.

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.

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
\[f(n,t) \gg \frac{t}{\log t}.\]

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 $x$?

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.

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 $\delta>0$) \[N(X,\delta)<X^{1/2+o(1)}?\]

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

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.

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 $N$?

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.

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 $n$.

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 $Q=\{3,5,7,11\}$?

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.

Is there a permutation $a_1,a_2,\ldots$ of the positive integers such that $a_k+a_{k+1}$ is always prime?

Asked by Segal. The answer is yes, as shown by Odlyzko.

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?

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?

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.

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 $n=4700063497$.

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.

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 $A\subset \mathbb{N}$ let
\[T(A) = \{ a_ix+b_i : 1\leq i\leq r\textrm{ and }x \in A\}.\]
Prove that, if $A_1=\{1\}$ and $A_{i+1}=T(A_i)$, then $\min(A_i) \ll 1$ for all $i\geq 0$.

Erdős and Graham write that 'it is surprising that [this problem] offers difficulty'.

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

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.

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.

See also [208].

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 $B=\{ N/2<p\leq N: p\textrm{ prime}\}$.

See also [425].

Let $A\subset \mathbb{C}$ be a finite set, 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 $A_k$ 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 $k\geq 0$).

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.

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].

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 \[2^{(1+o(1))\binom{n}{\lfloor n/2\rfloor}}.\]

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].

Let $M$ 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].

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 Turan. Turan 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].

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 $x\not\in f(y)$ 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.

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 $\leq \exp(n^{1-o(1)})$.

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 $2^{n-1}<N\leq 2^{n-1}+2^{n-3}$.

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 \[\alpha(n) \leq (\sqrt{3/2}+o(1))^{n}.\] The true order of magnitude may be $\alpha(n)=c^n$ for some $c>1$.

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]).

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 $1$.

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 $M(r)^\epsilon$)?

#514: \cit{Er61,p.249}

Boas (unpublished) has proved the first part, that such a path must exist.

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.

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 $\approx 0.7$.

Let $f$ be a random $\pm1$-valued multiplicative function, so that 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). Is it true that, almost surely,
\[\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N\log\log N}}=1?\]

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 $O(N^{1/2}/(\log\log N)^{5/2+o(1)})$.