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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.
Is there a covering system all of whose moduli are odd?
Asked by Erdős and Selfridge (sometimes also with Schinzel). They also asked whether there can be such that no two of the moduli divide each other, or where all the moduli are odd and squarefree?

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.

Additional thanks to: Antonio Girao
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 $\{1,\ldots,N\}$ (any number of the form $2^{2^m}-1$ with $m\geq 3$ suffices). Pan [Pa11] improved this to $\gg_\epsilon N^{1-\epsilon}$ for any $\epsilon>0$. Erdős believes this cannot be proved by covering systems, i.e. integers of the form $p+2^k+2^l$ exist in every infinite arithmetic progression.

See also [16].

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

Granville and Soundararajan [GrSo98] have conjectured that $3$ powers of 2 suffice.

See also [11].

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

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 $N$.
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.

Is the set of odd integers not of the form $2^k+p$ the union of an infinite arithmetic progression and a set of density $0$?
Erdős called this conjecture 'rather silly'. Using covering congruences Erdős [Er50] proved that the set of such odd integers contains an infinite arithmetic progression.

Chen [Ch23] has proved the answer is no.

See also [9].

Are there infinitely many primes $p$ such that every even number $n\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\leq p$?
The first prime without this property is $97$.
We call $m$ practical if every integer $n<m$ is the sum of distinct divisors of $m$. If $m$ is practical then let $h(m)$ be such that $h(m)$ many divisors always suffice.

Are there infinitely many practical $m$ such that \[h(m) < (\log\log m)^{O(1)}?\] Is it true that $h(n!)<n^{o(1)}$?

It is easy to see that almost all numbers are not practical. This may be true with $m=n!$. Erdős originally showed that $h(n!) <n$. Vose [Vo85] has shown that $h(n!)\ll n^{1/2}$.

Related to [304].

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-Lovász constant of $8/3$ has not been improved.
Additional thanks to: Zachary Chase
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 \[ \ll \frac{(\log\log n)^{3/2}}{(\log n)^{1/2}}n.\]
Additional thanks to: Mehtaab Sawhney
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.

See also the entry in the graphs problem collection.

Additional thanks to: Casey Tompkins
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].
Additional thanks to: Casey Tompkins, Tuan Tran
Let $n_1<n_2<\cdots$ be an arbitrary sequence of integers, each with an associated residue class $a_i\pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n<n_i$ or $n\not\equiv a_i\pmod{n_i}$. Must the logarithmic density of $A$ exist?
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 $a\in A$.
Given any infinite set $A\subset \mathbb{N}$ there is a set $B$ of density $0$ such that $A+B$ contains all except finitely many integers.
Proved by Lorentz [Lo54].
Is there a set $A\subset\mathbb{N}$ such that \[\lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2)\] and such that every large integer can be written as $p+a$ for some prime $p$ and $a\in A$? Can the bound $O(\log N)$ be achieved?
Erdős [Er54] proved that such a set $A$ exists with $\lvert A\cap\{1,\ldots,N\}\rvert\ll (\log N)^2$ (improving a previous result of Lorentz [Lo54] who achieved $\ll (\log N)^3$). Wolke [Wo96] has shown that such a bound is almost true, in that we can achieve $\ll (\log N)^{1+o(1)}$ if we only ask for almost all integers to be representable.
Let $A\subset\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\in A$ and $n\geq 0$. What is the smallest possible value of \[\limsup \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}?\]
Erdős observed that this value is finite and $>1$.
Additional thanks to: Timothy Gowers
Is there $A\subset\mathbb{N}$ such that every large integer can be written as $2^n+a$ for some $a\in A$ and $n\geq 0$ and \[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N/\log N}<\infty?\]
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 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 $A\subset\mathbb{N}$ 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 $A\subset\mathbb{N}$ which is not an additive basis, but is such that for every set $B\subseteq\mathbb{N}$ of density $\beta$ and every $N$ there exists $a\in A$ such that \[\frac{\lvert B\cap (B+a)\cap \{1,\ldots,N\}\rvert}{N}\geq (\beta+f(\beta))N\] where $f(\beta)>0$ for $0<\beta <1 $.
Let $M\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every maximal Sidon set $A\subset \{1,\ldots,N\}$ there is another Sidon set $B\subset \{1,\ldots,N\}$ of size $M$ such that $(A-A)\cap(B-B)=\{0\}$?
Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M=M(\epsilon)$ and $B\subset \{N+1,\ldots,M\}$ such that $A\cup B\subset \{1,\ldots,M\}$ is a Sidon set of size at least $(1-\epsilon)M^{1/2}$?
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 would be interesting - note that Croot's result allows for $n_k \leq e^{C^k}$ for some constant $C>1$. Is there a better bound?
Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$?
The answer is yes, as proved by Croot [Cr03].
Are there infinitely many integers $n,m$ such that $\phi(n)=\sigma(m)$?
This would follow immediately from the twin prime conjecture. The answer is yes, proved by Ford, Luca, and Pomerance [FLP10].
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 $t$.
Let $A$ be a finite set of integers. Is it true that, for every $k$, if $\lvert A\rvert$ is sufficiently large depending on $k$, then there are least $\lvert A\rvert^k$ many integers which are either the sum or product of distinct elements of $A$?
Asked by Erdős and Szemerédi [ErSz83]. Solved by Chang [Ch03].
Suppose $A\subseteq \{1,\ldots,N\}$ is such that there are no $k+1$ elements of $A$ which are relatively prime. An example is the set of all multiples of the first $k$ primes. Is this the largest such set?
This was disproved for $k=212$ by Ahlswede and Khachatrian [AhKh94], who suggest that their methods can disprove this for arbitrarily large $k$.

Erdős later asked [Er95] if the conjecture remains true provided $N\geq (1+o(1))p_k^2$ (or, in a weaker form, whether it is true for $N$ sufficiently large depending on $k$).

Additional thanks to: Zachary Chase
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.

Additional thanks to: David Penman
Is it true that the number of graphs on $n$ vertices which do not contain $G$ is \[\leq 2^{(1+o(1))\mathrm{ex}(n;G)}?\]
If $G$ is not bipartite the answer is yes, proved by Erdős, Frankl, and Rödl [ErFrRo86]. The answer is no for $G=C_6$, the cycle on 6 vertices. Morris and Saxton [MoSa16] have proved there are at least \[2^{(1+c)\mathrm{ex}(n;C_6)}\] such graphs for infinitely many $n$, for some constant $c>0$. It is still possible (and conjectured by Morris and Saxton) that the weaker bound of \[2^{O(\mathrm{ex}(n;G))}\] holds for all $G$.
Additional thanks to: Tuan Tran
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 $q$.

For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either a complete graph or independent set on $\geq n^c$ vertices?
Conjectured by Erdős and Hajnal [ErHa89], who proved that a complete graph or independent set must exist on \[\geq \exp(c_H\sqrt{\log n})\] many vertices, where $c_H>0$ is some constant. This was improved by Bucić, Nguyen, Scott, and Seymour [BNSS23] to \[\geq \exp(c_H\sqrt{\log n\log\log n}).\]
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.
Does every infinite graph with infinite chromatic number contain a cycle of length $2^n$ for infinitely many $n$?
Conjectured by Mihók and Erdős. It is likely that $2^n$ can be replaced by any sufficiently quickly growing sequence (e.g. the squares).
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?
A problem of Erdős and Hajnal. Gyárfás, Komlós, and Szemerédi [GyKoSz84] have proved that this sum is $\gg \log k$. Liu and Montgomery [LiMo20] have proved the asymptotically sharp lower bound of $\geq (\tfrac{1}{2}-o(1))\log k$.

See also the entry in the graphs problem collection.

Is \[\sum_{n\geq 2}\frac{1}{n!-1}\] irrational?
Is \[\sum_{n\geq 2}\frac{\omega(n)}{2^n}\] irrational? (Here $\omega(n)$ counts the number of distinct prime divisors of $n$.)
Erdős [Er48] proved that $\sum_n \frac{d(n)}{2^n}$ is irrational, where $d(n)$ is the divisor function.
Is every integer $n\equiv 0\pmod{4}$ equal to $2^k+m$ for some $k\geq 0$ and squarefree integer $m$?
Erdős could prove this is true for almost all $n\equiv 0\pmod{4}$.
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.
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.)
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].

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

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

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

See also the entry in the graphs problem collection.

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

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

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 $n^{1/2}(\log n)^{O(1)}$.
Additional thanks to: Zachary Hunter
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 $c>0$.
Additional thanks to: Tuan Tran
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 $C_4$.
Let $\epsilon >0$. Is it true that, if $k$ is sufficiently large, then \[R(G)>(1-\epsilon)^kR(k)\] for every graph $G$ with chromatic number $\chi(G)=k$?

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

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

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

See also the entry in the graphs problem collection and this second one.

Additional thanks to: Yuval Wigderson
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.
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?

Additional thanks to: Boris Alexeev and Dustin Mixon
Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\to \infty$?
Erdős could not even prove $h(n)\geq n$. Pach has shown $h(n)<n^{\log_23}$.
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 $<5$).
Additional thanks to: Shengtong Zhang
Given $n$ points in $\mathbb{R}^2$ such that there are $\geq cn^2$ lines each containing more than three points there must be a single line containing $\gg_c n^{1/2}$ many points.
Even replacing $n^{1/2}$ with any function $\to\infty$, this conjecture is open.
Let $A$ be a set of $n$ points in $\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\gg n$?
Perhaps the diameter is even $\geq n-1$ for sufficiently large $n$. Kanold proved the diameter is $\geq n^{3/4}$. The bounds on the distinct distance problem [89] proved by Guth and Katz [GuKa15] imply a lower bound of $\gg n/\log n$.
Additional thanks to: Boris Alexeev and Dustin Mixon
Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.
In [Er81d] Erdős proved that $\gg n$ many circles is possible, and that there cannot be more than $n(n-1)$ many circles. Elekes [El84] has a simple construction of a set with $\gg n^{3/2}$ such circles. This may be the correct order of magnitude.
Draw $n$ squares inside the unit square with no common interior point. Let $f(n)$ be the maximum possible total perimeter of the squares. Is $f(k^2+1)=4k$?
It is trivial from the Cauchy-Schwarz inequality that $f(k^2)=4k$. Erdős also asks for which $n$ is it true that $f(n+1)=f(n)$.
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].
Additional thanks to: Antonio Girao
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].
Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a directed path of size $m$. Determine $k(n,m)$.
A problem of Erdős and Rado [ErRa67], who showed \[k(n,m) \leq \frac{2^{m-1}(n-1)^m+n-2}{2n-3}.\] Larson and Mitchell [LaMi97] prove that $k(n,m)\leq n^{m-1}$ for $m\geq 3$.

See also the entry in the graphs problem collection.

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 $p(z)=z^n-1$?
Additional thanks to: Geoffrey Irving
If $p(z)$ is a polynomial of degree $n$ such that $\{z : \lvert p(z)\rvert\leq 1\}$ is connected then is it true that \[\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?\]
The lower bound is easy: this is $\geq n$ and equality holds if and only if $p(z)=z^n$. The assumption that the set is connected is necessary, as witnessed for example by $p(z)=z^2+10z+1$.

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

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

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

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

Additional thanks to: Boris Alexeev and Dustin Mixon
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.
Additional thanks to: Zachary Chase
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 $\sigma(n)$.
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 $\{3,4,7\}$.
Additional thanks to: Boris Alexeev and Dustin Mixon
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].
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.
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.
Additional thanks to: Antonio Girao
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 $C$.
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.
Let $k>3$. Does the product of any $k$ consecutive integers $N=\prod_{m< n\leq m+k}n$ (with $m>k$) have a prime factor $p\mid N$ such that $p^2\nmid N$?
Erdős and Selfridge proved that $N$ can never be a perfect power. Erdős remarked that this 'seems hopeless at present'.
Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$-coloured there must exist a monochromatic $k$-term arithmetic progression. Improve the bounds for $W(k)$ - for example, prove that $W(k)^{1/k}\to \infty$.
When $p$ is prime Berlekamp [Be68] has proved $W(p+1)\geq p2^p$. Gowers [Go01] has proved \[W(k) \leq 2^{2^{2^{2^{2^{k+9}}}}}.\]
Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression?
Green and Tao [GrTa08] have proved that there must always exist some $k$ primes in arithmetic progression, but these need not be consecutive. Erdős called this conjecture 'completely hopeless at present'.
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 $F(k)$ be the number of solutions to \[ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},\] where $1\leq n_1<\cdots<n_k$ are distinct integers. Find good estimates for $F(k)$.
The current best bounds known are \[2^{c^{\frac{k}{\log k}}}\leq F(k) \leq c_0^{(\frac{1}{5}+o(1))2^k},\] where $c>0$ is some absolute constant and $c_0=1.26408\cdots$ is the 'Vardi constant'. The lower bound is due to Konyagin [Ko14] and the upper bound to Elsholtz and Planitzer [ElPl21].
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 maximal clique (on at least $2$ vertices) in $G$ contains at least one vertex from $X$. Let $H(n)$ be maximal such that every triangle-free graph on $n$ vertices contains an independent set on $H(n)$ vertices. Does $h(n)=n-H(n)$?
It is easy to see that $h(n)\leq n-\sqrt{n}$ and that $h(n)\leq n-H(n)$. Conjectured by Erdős and Gallai, who were unable to make progress even assuming $G$ is $K_4$-free. Erdős remarked that this conjecture is 'perhaps completely wrongheaded'.
Additional thanks to: Zachary Chase
For any $M\geq 1$, if $A\subset \mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\in A+A$ such that $a+1,a-1\not\in A+A$.
There may even be $\gg \lvert A\rvert^2$ many such $a$. A similar question can be asked for truncations of infinite Sidon sets.
Additional thanks to: Cedric Pilatte
Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$. Is it true that \[\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty\] as $\lvert A\rvert\to \infty$?
A similar problem can be asked for infinite Sidon sets.
Let $A\subset \{1,\ldots,N\}$ be a Sidon set with $\lvert A\rvert\sim N^{1/2}$. Must $A+A$ be well-distributed over all small moduli? In particular, must about half the elements of $A+A$ be even and half odd?
Lindström [Li98] has shown this is true for $A$ itself, subsequently strengthened by Kolountzakis [Ko99].
Let $F(N)$ be the size of the largest Sidon subset of $\{1,\ldots,N\}$. Is it true that for every $k\geq 1$ we have \[F(N+k)\leq F(N)+1\] for all sufficiently large $N$?
This may even hold with $k\approx \epsilon N^{1/2}$.
Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$?
Does there exist an infinite Sidon set which is an asymptotic basis of order 3?
Yes, as shown by Pilatte [Pi23].
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.
There exists some constant $c>0$ such that $$R(C_4,K_n) \ll n^{2-c}.$$
The current bounds are \[ \frac{n^{3/2}}{(\log n)^{3/2}}\ll R(C_4,K_n)\ll \frac{n^2}{(\log n)^2}.\] The upper bound is due to Szemerédi (mentioned in [EFRS78]), and the lower bound is due to Spencer [Sp77].

See also the entry in the graphs problem collection.

Let $h(N)$ be the smallest $k$ such that $\{1,\ldots,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate $h(N)$.
Investigated by Erdős and Freud. This has been discussed on MathOverflow, where LeechLattice shows \[h(N) \ll N^{2/3}.\] The observation of Zachary Hunter in that question coupled with the bounds of Kelley-Meka [KeMe23] imply that \[h(N) \gg \exp(c(\log N)^{1/12})\] for some $c>0$.
Additional thanks to: Zachary Hunter
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. This is equivalent to showing that if $H$ is the union of $c$ forests then $R(H)\ll_c n$, and also that if every subgraph has average degree at most $d$ then $R(H)\ll_d n$. Solved by Lee [Le16], who proved that \[ R(H) \leq 2^{2^{O(d)}}n.\]

See also the entry in the graphs problem collection.

A set $A\subset \mathbb{N}$ is primitive if no member of $A$ divides another. Is the sum \[\sum_{n\in A}\frac{1}{n\log n}\] maximised over all primitive sets when $A$ is the set of primes?
Erdős [Er35] proved that this sum always converges for a primitive set. Solved by Lichtman [Li23].
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.
Additional thanks to: Jonathan Chapman
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 for every $\epsilon>0$ and integer $t\geq 1$, if $N$ is sufficiently large and $A$ is a subset of $[t]^N$ of size at least $\epsilon t^N$ then $A$ must contain a combinatorial line $P$ (a set $P=\{p_1,\ldots,p_t\}$ where for each coordinate $1\leq j\leq t$ the $j$th coordinate of $p_i$ is either $i$ or constant).
The 'density Hales-Jewett' problem. This was proved by Furstenberg and Katznelson [FuKa91]. A new elementary proof, which gives quantitative bounds, was proved by the Polymath project [Po09].
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 $\lvert A\rvert=2$.

Additional thanks to: Ryan Alweiss
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 $A_1,A_2,\ldots$ be an infinite collection of infinite sets of integers, say $A_i=\{a_{i1}<a_{i2}<\cdots\}$. Does there exist some $f:\mathbb{N}\to\{-1,1\}$ such that \[\max_{m, 1\leq i\leq d} \left\lvert \sum_{1\leq j\leq m} f(a_{ij})\right\rvert \ll_d 1\] for all $d\geq 1$?
Erdős remarks 'it seems certain that the answer is affirmative'. This was solved by Beck [Be81]. Recently Beck [Be17] proved that one can replace $\ll_d 1$ with $\ll d^{4+\epsilon}$ for any $\epsilon>0$.
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 $G\in\mathcal{F}$ such that \[\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?\]

A problem of Erdős and Simonovits.

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

See also [575] and the entry in the graphs problem collection.

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.)

See also the entry in the graphs problem collection.

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.

Additional thanks to: Antonio Girao
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.

See also [583].

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 $\ll (N\log N)^{1/2}$).
Additional thanks to: Zachary Chase
Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference $d$ of length $f(d)$ for infinitely many $d$.
Originally asked by Cohen. Erdős observed that colouring according to whether $\{ \sqrt{2}n\}<1/2$ or not implies $f(d) \ll d$ (using the fact that $\|\sqrt{2}q\| \gg 1/q$ for all $q$, where $\|x\|$ is the distance to the nearest integer). Beck [Be80] has improved this using the probabilistic method, constructing a colouring that shows $f(d)\leq (1+o(1))\log_2 d$. Van der Waerden's theorem implies $f(d)\to \infty$ is necessary.
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.

Additional thanks to: Ryan Alweiss, Vjekoslav Kovac
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 \[\sum_{x\in X}\frac{1}{\log x}\geq 2^{-8}\log\log\log n.\]

Additional thanks to: Mehtaab Sawhney
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.
Additional thanks to: Boris Alexeev and Dustin Mixon
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].

See also [195] and [196].

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

See also [194] and [196].

Additional thanks to: Boris Alexeev and Dustin Mixon
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.

See also [194] and [195].

Additional thanks to: Boris Alexeev and Dustin Mixon
Can $\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?
If three sets are allowed then this is possible.
Additional thanks to: Boris Alexeev and Dustin Mixon
If $A\subset \mathbb{N}$ is a Sidon set then must the complement of $A$ contain an infinite arithmetic progression?
The answer is yes, as shown by Baumgartner [Ba75].
If $A\subset \mathbb{R}$ does not contain a 3-term arithmetic progression then must $\mathbb{R}\backslash A$ contain an infinite arithmetic progression?
The answer is no, as shown by Baumgartner [Ba75] (whose construction uses the axiom of choice).
Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$?
It follows from the prime number theorem that such a progression has length $\leq(1+o(1))\log N$.
Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\{1,\ldots,N\}$ without a $k$-term arithmetic progression? Is it true that \[\lim_{N\to \infty}\frac{R_3(N)}{G_3(N)}=1?\]
First asked and investigated by Riddell [Ri69]. It is trivial that $G_k(N)\leq R_k(N)$, and it is possible that $G_k(N) <R_k(N)$ (for example with $k=3$ and $N=14$). Komlós, Sulyok, and Szemerédi [KSS75] have shown that $R_k(N) \ll_k G_k(N)$.
Additional thanks to: Zachary Chase
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.
Let $x\in \overline{\mathbb{Q}}$ be an algebraic number. For any $n\geq 1$ let \[R_n(x) = \sum_{i=1}^n\frac{1}{m_i}<x\] be the maximal sum of $n$ distinct unit fractions which is $<x$. Is it true that, for sufficiently large $n$, we have \[R_{n+1}(x)=R_n(x)+\frac{1}{m},\] where $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is $<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?)
If $x$ is irrational then this can be false, but it may be true for almost all $x\in\mathbb{R}$. Curtiss [Cu22] showed that this is true for $x=1$ and Erdős [Er50b] showed it is true for all $x\in\mathbb{Q}$.
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 $d\geq 4$.
Additional thanks to: Juan Escudero
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 $n$.
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.
Additional thanks to: Bhavik Mehta
Does there exist $S\subseteq \mathbb{R}^2$ such that every set congruent to $S$ (that is, $S$ after some translation and rotation) contains exactly one point from $\mathbb{Z}^2$?
An old question of Steinhaus. Erdős was 'almost certain that such a set does not exist'.

In fact, such a set does exist, as proved by Jackson and Mauldin [JaMa02]. Their construction depends on the axiom of choice.

Additional thanks to: Vjekoslav Kovac
Let $g(k)$ be the smallest integer (if any such exists) such that any $g(k)$ points in $\mathbb{R}^2$ contains an empty convex $k$-gon (i.e. with no point in the interior). Does $g(k)$ exist? If so, estimate $g(k)$.
A variant of the 'happy ending' problem, 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 $n\geq 7$.

See also the entry in the graphs problem collection.

Additional thanks to: Zachary Hunter
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 $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.
Are there arbitrarily long arithmetic progressions of primes?
The answer is yes, proved by Green and Tao [GrTa08]. The stronger claim that there are arbitrarily long arithmetic progressions of consecutive primes is still open.
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] proved there is such a set with, for all large $N$, \[\lvert A\cap\{1,\ldots,N\}\rvert \ll \frac{\log\log N}{\log N}N\]
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 \[n_{k+1}-n_k \ll n_k^{1/4}.\]
Let $d\geq 2$ and $n\geq 2$. Let $f_d(n)$ be maximal such that, for any $A\subseteq \mathbb{R}^d$ of size $n$, with diameter $1$, the distance 1 occurs between $f_d(n)$ many pairs of points in $A$. Estimate $f_d(n)$.
Hopf and Pannwitz [HoPa34] proved $f_2(n)=n$. Heppes [He56] and Grünbaum-Strasziewicz independently showed that $f_3(n)=2n-2$.
If $A\subseteq \mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determine an obtuse angle.
For $d=2$ this is trivial. For $d=3$ there is an unpublished proof by Kuiper and Boerdijk. The general case was proved by Danzer and Grünbaum [DaGr62].
Additional thanks to: Boris Alexeev and Dustin Mixon
Let \[ f(\theta) = \sum_{k\geq 1}c_k e^{ik\theta}\] be a trigonometric polynomial (so that the $c_k\in \mathbb{C}$ are finitely supported) with real roots such that $\max_{\theta\in [0,2\pi]}\lvert f(\theta)\rvert=1$. Then \[\int_0^{2\pi}\lvert f(\theta)\rvert \mathrm{d}\theta \leq 4.\]
This was solved independently by Kristiansen [Kr74] (only in the case when $c_k\in\mathbb{R}$) and Saff and Sheil-Small [SSS73] (for general $c_k\in \mathbb{C}$).
Additional thanks to: Winston Heap, Vjekoslav Kovac, Karlo Lelas
Is there an entire non-linear function $f$ such that, for all $x\in\mathbb{R}$, $x$ is rational if and only if $f(x)$ is?
More generally, if $A,B\subseteq \mathbb{R}$ are two countable dense sets then is there an entire function such that $f(A)=B$?
Additional thanks to: Boris Alexeev and Dustin Mixon
Let $f=\sum_{n=0}^\infty a_nz^n$ be an entire function. Is it true that if \[\lim_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}\] exists then it must be $0$?
Clunie (unpublished) proved this if $a_n\geq 0$ for all $n$. This was disproved in general by Clunie and Hayman [ClHa64], who showed that the limit can take any value in $[0,1/2]$.

See also [513].

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].

Additional thanks to: Mehtaab Sawhney
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 \[f^{(k_n)}(z) = 0\textrm{ for all }z\in S_n?\]
Additional thanks to: Zachary Chase
Let $P(z)=\sum_{1\leq k\leq n}a_kz^k$ for some $a_k\in \mathbb{C}$ with $\lvert a_k\rvert=1$ for $1\leq k\leq n$. Does there exist a constant $c>0$ such that, for $n\geq 2$, we have \[\max_{\lvert z\rvert=1}\lvert P(z)\rvert \geq (1+c)\sqrt{n}?\]
The lower bound of $\sqrt{n}$ is trivial from Parseval's theorem. The answer is no (contrary to Erdős' initial guess). Kahane [Ka80] constructed 'ultraflat' polynomials $P(z)=\sum a_kz^k$ with $\lvert a_k\rvert=1$ such that \[P(z)=(1+o(1))\sqrt{n}\] uniformly for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$, where the $o(1)$ term $\to 0$ as $n\to \infty$.

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

See also [228].

Additional thanks to: Mehtaab Sawhney
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 $d_n=p_{n+1}-p_n$. Prove that \[\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.\]
Cramer proved an upper bound of $O(N(\log N)^4)$ conditional on the Riemann hypothesis. The prime number theorem immediately implies a lower bound of $\gg N(\log N)^2$.
For every $c\geq 0$ the density $f(c)$ of integers for which \[\frac{p_{n+1}-p_n}{\log n}< c\] exists and is a continuous function of $c$.
Let $N_k=2\cdot 3\cdots p_k$ and $\{a_1<a_2<\cdots <a_{\phi(N_k)}\}$ be the integers $<N_k$ which are relatively prime to $N_k$. Then, for any $c\geq 0$, the limit \[\frac{\#\{ a_i-a_{i-1}\leq c \frac{N_k}{\phi(N_k)} : 2\leq i\leq \phi(N_k)\}}{\phi(N_k)}\] exists and is a continuous function of $c$.
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 $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap \{1,\ldots,N\}\rvert \gg \log N$ for all large $N$. Let $f(n)$ count the number of solutions to $n=p+a$ for $p$ prime and $a\in A$. Is it true that $\limsup f(n)=\infty$?
Erdős [Er50] proved this when $A=\{2^k : k\geq 0\}$. Solved by Chen and Ding [ChDi22].
Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\log x$ many consecutive primes $\leq x$ such that the difference between any two is $>c_2$?
This is well-known if $c_1$ is sufficiently small.
Let $f:\mathbb{N}\to \{-1,1\}$ be a multiplicative function. Is it true that \[ \lim_{N\to \infty}\frac{1}{N}\sum_{n\leq N}f(n)\] always exists?
Wintner observed that if $f$ can take complex values on the unit circle then the limit need not exist. The answer is yes, as proved by Wirsing [Wi67], and generalised by Halász [Ha68].
Is there an infinite set of primes $P$ such that if $\{a_1<a_2<\cdots\}$ is the set of integers divisible only by primes in $P$ then $\lim a_{i+1}-a_i=\infty$?
Originally asked to Erdős by Wintner. The limit is infinite for a finite set of primes, which follows from a theorem of Pólya.
Additional thanks to: Boris Alexeev and Dustin Mixon
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 $C>1$. Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?
Originally asked to Erdős by Kalmár. Erdős believed the answer is yes. Romanoff [Ro34] proved that the answer is yes if $C$ is an integer.
Let $A\subseteq \mathbb{N}$ be an infinite set such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that \[\limsup_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}\geq 3?\]
Erdős writes it is 'easy to see' that this holds with $3$ replaced by $2$, and that $3$ would be best possible here. We do not see an easy argument that this holds with $2$, but this follows e.g. from the main result of Mann [Ma60].

The answer is yes, proved by Freiman [Fr73].

Additional thanks to: Zachary Chase
Let $(a,b)=1$. Every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\geq 0$.
Proved by Birch [Bi59].
Let $n_1<n_2<\cdots$ be a sequence of integers such that \[\limsup \frac{n_k}{k}=\infty.\] Is \[\sum_{k=1}^\infty \frac{1}{2^{n_k}}\] transcendental?
Erdős [Er75c] proved the answer is yes under the stronger condition that $\limsup n_k/k^t=\infty$ for all $t\geq 1$.
Are there infinitely many $n$ such that, for all $k\geq 1$, \[ \omega(n+k) \ll k,\] where the implied constant depends only on $n$? (Here $\omega(n)$ is the number of distinct prime divisors of $n$.)
Related to [69]. Erdős and Graham [ErGr80] write 'we just know too little about sieves to be able to handle such a question ("we" here means not just us but the collective wisdom (?) of our poor struggling human race).'
Is \[\sum_n \frac{\phi(n)}{2^n}\] irrational? Here $\phi$ is the Euler totient function.
Is \[\sum \frac{\sigma(n)}{2^n}\] irrational? (Here $\sigma(n)$ is the sum of divisors function.)
The answer is yes, as shown by Nesterenko [Ne96].
Is \[\sum \frac{p_n}{2^n}\] irrational? (Here $p_n$ is the $n$th prime.)
Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$. Is \[\sum \frac{\sigma_k(n)}{n!}\] irrational?
This is known now for $1\leq k\leq 4$. The cases $k=1,2$ are reasonably straightforward, as observed by Erdős [Er52]. The case $k=3$ was proved independently by Schlage-Puchta [ScPu06] and Friedlander, Luca, and Stoiciu [FLC07]. The case $k=4$ was proved by Pratt [Pr22].
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 $z_1,z_2,\ldots \in [0,1]$ be an infinite sequence, and define the discrepancy \[D_N(I) = \#\{ n\leq N : z_n\in I\} - N\lvert I\rvert.\] Must there exist some interval $I\subseteq [0,1]$ such that \[\limsup_{N\to \infty}\lvert D_N(I)\rvert =\infty?\]
The answer is yes, as proved by Schmidt [Sc68], who later showed [Sc72] that in fact this is true for all but countably many intervals of the shape $[0,x]$.

Essentially the best possible result was proved by Tijdeman and Wagner [TiWa80], who proved that, for almost all intervals of the shape $[0,x)$, we have \[\limsup_{N\to \infty}\frac{\lvert D_N([0,x))\rvert}{\log N}\gg 1.\]

Additional thanks to: Cedric Pilatte and Stefan Steinerberger
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 $c>0$.
Additional thanks to: Zachary Chase
Let $A\subseteq \mathbb{N}$ be an infinite set. Is \[\sum_{n\in A}\frac{1}{2^n-1}\] irrational?
If $A=\mathbb{N}$ then this series is $\sum_{n}\frac{d(n)}{2^n}$, where $d(n)$ is the number of divisors of $n$, which is known to be irrational.
Let $a_n\to \infty$. Is \[\sum_{n} \frac{d(n)}{a_1\cdots a_n}\] irrational, where $d(n)$ is the number of divisors of $n$?
Erdős and Straus [ErSt71] have proved this is true if $a_n$ is monotone, i.e. $a_{n-1}\leq a_n$ for all $n$.
Is the sum \[\sum_{n} \mu(n)^2\frac{n}{2^n}\] irrational?
Additional thanks to: Boris Alexeev and Dustin Mixon
Let $a_n$ be a sequence such that $a_n/n\to \infty$. Is the sum \[\sum_n \frac{a_n}{2^{a_n}}\] irrational?
This is true under either of the stronger assumptions that
  • $a_{n+1}-a_n\to \infty$ or
  • $a_n \gg n\sqrt{\log n\log\log n}$.
Erdős and Graham speculate that the condition $\limsup a_{n+1}-a_n=\infty$ is not sufficient, but know of no example.
Are there infinitely many $n$ such that there exists some $t\geq 2$ and $a_1,\ldots,a_t\geq 1$ such that \[\frac{n}{2^n}=\sum_{1\leq k\leq t}\frac{a_k}{2^{a_k}}?\] Is this true for all $n$? Is there a rational $x$ such that \[x = \sum_{k=1}^\infty \frac{a_k}{2^{a_k}}\] has at least two solutions?
Related to [260].
Additional thanks to: Zachary Chase
Suppose $a_1<a_2<\cdots$ is a sequence of integers such that for all integer sequences $t_n$ with $t_n\geq 1$ the sum \[\sum_{n=1}^\infty \frac{1}{t_na_n}\] is irrational. How slowly can $a_n$ grow?
One possible definition of an 'irrationality sequence' (see also [263] and [264]). An example of such a sequence is $a_n=2^{2^n}$, while a non-example is $a_n=n!$. It is known that if $a_n$ is such a sequence then $a_n^{1/n}\to\infty$.
Let $a_n$ be a sequence of integers such that for every sequence of integers $b_n$ with $b_n/a_n\to 1$ the sum \[\sum\frac{1}{b_n}\] is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\to \infty$?
One possible definition of an 'irrationality sequence' (see also [262] and [264]).
Let $a_n$ be a sequence of integers such that, for every bounded sequence $b_n$, the sum \[\sum \frac{1}{a_n+b_n}\] is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence? Is there such a sequence with $a_n<n^k$?
A possible definition of an 'irrationality sequence' (see also [262] and [263]). One example is $a_n=2^{2^n}$.
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 $a_n$ be an infinite sequence of integers. There exists some integer $t\geq 1$ such that \[\sum \frac{1}{a_n+t}\] is irrational.
This conjecture is due to Stolarsky.
Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_1<n_2<\cdots $ be an infinite sequence with $n_{k+1}/n_k \geq c>1$. Must \[\sum_k\frac{1}{F_{n_k}}\] be irrational?
It may be sufficient to have $n_k/k\to \infty$. Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that $\sum \frac{1}{F_{2^n}}$ is irrational. The nature of $\sum \frac{1}{F_n}$ itself is unknown.
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.

Additional thanks to: Zachary Hunter
Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\geq 5$?
Selfridge has found an example using divisors of $360$ if $p=3$ is allowed.
If $G$ is an abelian group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)
A problem of Herzog and Schönheim.
Additional thanks to: Boris Alexeev and Dustin Mixon
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'.
For which $n$ is there a covering system $\{ a_d\pmod{d} : d\mid n\}$ such that if $x\equiv a_{d_1}\pmod{d_1}$ and $x\equiv a_{d_2}\pmod{d_2}$ then $(d_1,d_2)=1$?
The density of such $n$ is zero.
Let $A=\{n_1<\cdots<n_r\}$ be a finite set of integers. What is the minimum value of the density of integers not hit by a suitable choice of congreunces $a_i\pmod{n_i}$? Is the minimum achieved when all the $a_i$ are equal?
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.

Additional thanks to: Zachary Hunter
Let $p:\mathbb{Z}\to \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\geq 2$ with $d\mid p(n)$ for all $n\geq 1$. Is it true that, for all sufficiently large $m$, there exist integers $1\leq n_1<\cdots <n_k$ such that \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\] and \[m=p(n_1)+\cdots+p(n_k)?\]
Graham [Gr63] has proved this when $p(x)=x$. Cassels [Ca60] has proved that these conditions on the polynomial imply every sufficiently large integer is the sum of $p(n_i)$ with distinct $n_i$. Burr has proved this if $p(x)=x^k$ with $k\geq 1$ and if we allow $n_i=n_j$.
Let $f(k)$ be the maximal value of $n_1$ such that there exist $n_1<n_2<\cdots <n_k$ with \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\] Is it true that \[f(k)=(1+o(1))\frac{k}{e-1}?\]
The upper bound $f(k) \leq (1+o(1))\frac{k}{e-1}$ is trivial since for any $u\geq 1$ we have \[\sum_{u\leq n\leq eu}\frac{1}{n}=1+o(1),\] and hence if $f(k)=u$ then we must have $k\geq (e-1-o(1))u$. Essentially solved by Croot [Cr01], who showed that for any $N>1$ there exists some $k\geq 1$ and \[N<n_1<\cdots <n_k \leq (e+o(1))N\] with $1=\sum \frac{1}{n_i}$.
Additional thanks to: Zachary Hunter
Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1<n_2<\cdots <n_k$ with \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\] Is it true that \[f(k)=(1+o(1))\frac{e}{e-1}k?\]
It is trivial that $f(k)\geq (1+o(1))\frac{e}{e-1}k$, since for any $u\geq 1$ \[\sum_{e\leq n\leq eu}\frac{1}{n}= 1+o(1),\] and so if $eu\approx f(k)$ then $k\leq \frac{e-1}{e}f(k)$. Proved by Martin [Ma00].
Additional thanks to: Zachary Hunter
Let $k\geq 2$. Is it true that there exists an interval $I$ of width $(e-1+o(1))k$ and integers $n_1<\cdots<n_k\in I$ such that \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?\]
The answer is yes, proved by Croot [Cr01].
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.

Is it true that there are only finitely many pairs of intervals $I_1,I_2$ such that \[\sum_{n_1\in I_1}\frac{1}{n_1}+\sum_{n_2\in I_2}\frac{1}{n_2}\in \mathbb{N}?\]
This is still open even if $\lvert I_2\rvert=1$. It is perhaps true with two intervals replaced by any $k$ intervals.
Is it true that, for all sufficiently large $k$, there exist intervals $I_1,\ldots,I_k$ with $\lvert I_i\rvert \geq 2$ for $1\leq i\leq k$ such that \[1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}?\]
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 $n\geq 1$ and define $L_n$ to be the lowest common multiple of $\{1,\ldots,n\}$ and $a_n$ by \[\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}.\] Is it true that $(a_n,L_n)=1$ and $(a_n,L_n)>1$ both occur for infinitely many $n$?
Let $A$ be the set of $n\in \mathbb{N}$ such that there exist $1\leq m_1<\cdots <m_k=n$ with $\sum\tfrac{1}{m_i}=1$. Explore $A$. In particular,
  • Does $A$ have density $1$?
  • What are those $n\in A$ not divisible by any $d\in A$ with $1<d<n$?
Straus observed that $A$ is closed under multiplication. Furthermore, it is easy to see that $A$ does not contain any prime power.
Additional thanks to: Zachary Hunter
Let $k\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\] with $1\leq n_1<\cdots <n_k$. Estimate the growth of $v(k)$.
Results of Bleicher and Erdős [BlEr75] imply $v(k) \gg k!$. It may be that $v(k)$ grows doubly exponentially in $\sqrt{k}$ or even $k$.

An elementary inductive argument shows that $n_k\leq ku_k$ where $u_1=1$ and $u_{i+1}=u_i(u_i+1)$, and hence \[v(k) \leq kc_0^{2^k},\] where \[c_0=\lim_n u_n^{1/2^n}=1.26408\cdots\] is the 'Vardi constant'.

Additional thanks to: Zachary Hunter
Let $N\geq 1$ and let $t(N)$ be the least integer $t$ such that there is no solution to \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\] with $t=n_1<\cdots <n_k\leq N$. Estimate $t(N)$.
Erdős and Graham [ErGr80] could show \[t(N)\ll\frac{N}{\log N},\] but had no idea of the true value of $t(N)$.

Solved by Liu and Sawhney [LiSa24] (up to $(\log\log N)^{O(1)}$), who proved that \[\frac{N}{(\log N)(\log\log N)^3(\log\log\log N)^{O(1)}}\ll t(N) \ll \frac{N}{\log N}.\]

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

Additional thanks to: Zachary Hunter and Mehtaab Sawhney
Let $N\geq 1$. How many $A\subseteq \{1,\ldots,N\}$ are there such that $\sum_{n\in A}\frac{1}{n}=1$?
It was not even known for a long time whether this is $2^{cN}$ for some $c<1$ or $2^{(1+o(1))N}$.

Steinerberger [St24] has shown that in fact the number of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in A}\frac{1}{n}\leq 1$ is at most $2^{0.93N}$. Independently, Liu and Sawhney [LiSa24] gave a best possible answer, proving that the number of $A\subseteq \{1,\ldots,N\}$ with $\sum_{n\in A}\frac{1}{n}=1$ is \[(1.88057\cdots )^{(1+o(1))n},\] where $1.88057\cdots$ is an explicit number defined as the solution to a certain integral equation.

Does every set $A\subseteq \mathbb{N}$ of positive density contain some finite $S\subset A$ such that $\sum_{n\in S}\frac{1}{n}=1$?
The answer is yes, proved by Bloom [Bl21].
Is there an infinite sequence $a_1<a_2<\cdots $ such that $a_{i+1}-a_i=O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to $1$?
There does not exist such a sequence, which follows from the positive solution to [298] by Bloom [Bl21].
Let $A(N)$ denote the maximal cardinality of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in S}\frac{1}{n}\neq 1$ for all $S\subseteq A$. Estimate $A(N)$.
Erdős and Graham believe the answer is $A(N)=(1+o(1))N$. Croot [Cr03] disproved this, showing the existence of some constant $c<1$ such that $A(N)<cN$ for all large $N$. It is trivial that $A(N)\geq (1-\frac{1}{e}+o(1))N$.

Liu and Sawhney [LiSa24] have proved that $A(N)=(1-1/e+o(1))N$.

Additional thanks to: Zachary Hunter
What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that for any $a,b_1,\ldots,b_k\in A$ with $k\geq 2$ we have \[\frac{1}{a}\neq \frac{1}{b_1}+\cdots+\frac{1}{b_k}?\] Is there a constant $c>1/2$ such that $\lvert A\rvert >cN$ is possible for all large $N$?
The example $A=(N/2,N]\cap \mathbb{N}$ shows that $\lvert A\rvert\geq N/2$.
Additional thanks to: Zachary Hunter
Is it true that, for any $\delta>1/2$, if $N$ is sufficiently large and $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert \geq \delta N$ then there exist $a,b,c\in A$ such that \[\frac{1}{a}=\frac{1}{b}+\frac{1}{c}.\]
The colouring version of this is [303], which was solved by Brown and Rödl [BrRo91].

The possible alternative question, that if $A\subseteq \mathbb{N}$ is a set of positive lower density then must there exist $a,b,c\in A$ such that \[\frac{1}{a}=\frac{1}{b}+\frac{1}{c},\] has a negative answer, taking for example $A$ to be the union of $[5^k,(1+\epsilon)5^k]$ for large $k$ and sufficiently small $\epsilon>0$. This was observed by Hunter and Sawhney.

Additional thanks to: Zachary Hunter and Mehtaab Sawhney
Is it true that in any finite colouring of the integers there exists a monochromatic solution to \[\frac{1}{a}=\frac{1}{b}+\frac{1}{c}\] with distinct $a,b,c$?
The density version of this is [302]. This colouring version is true, as proved by Brown and Rödl [BrRo91].
For integers $1\leq a<b$ let $N(a,b)$ denote the minimal $k$ such that there exist integers $1<n_1<\cdots<n_k$ with \[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\] Estimate $N(b)=\max_{1\leq a<b}N(a,b)$. Is it true that $N(b) \ll \log\log b$?
Erdős [Er50c] proved that \[\log\log b \ll N(b) \ll \frac{\log b}{\log\log b}.\] The upper bound was improved by Vose [Vo85] to \[N(b) \ll \sqrt{\log b}.\] One can also investigate the average of $N(a,b)$ for fixed $b$, and it is known that \[\frac{1}{b}\sum_{1\leq a<b}N(a,b) \gg \log\log b.\]

Related to [18].

Additional thanks to: Zachary Hunter
For integers $1\leq a<b$ let $D(a,b)$ be the minimal value of $n_k$ such that there exist integers $1\leq n_1<\cdots <n_k$ with \[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\] Estimate $D(b)=\max_{1\leq a<b}D(a,b)$. Is it true that \[D(b) \ll b(\log b)^{1+o(1)}?\]
Bleicher and Erdős [BlEr76] have shown that \[D(b)\ll b(\log b)^2.\] If $b=p$ is a prime then \[D(p) \gg p\log p.\]

This was solved by Yokota [Yo88], who proved that \[D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2.\] This was improved by Liu and Sawhney [LiSa24] to \[D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}.\]

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 $N\geq 1$. What is the smallest integer not representable as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$? Is it true that the set of integers not representable as such has the shape $[m,\infty)$ for some $m$?
Let $N\geq 1$. How many integers can be written as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$? Are there $o(\log N)$ such integers?
The answer to the second question is no: there are at least $(1-o(1))\log N$ many such integers, which follows from a more precise result of Croot [Cr99], who showed that every integer at most \[\leq \sum_{n\leq N}\frac{1}{n}-(\tfrac{9}{2}+o(1))\frac{(\log\log N)^2}{\log N}\] can be so represented.
Additional thanks to: Zachary Hunter and Mehtaab Sawhney
Let $\alpha >0$ and $N\geq 1$. Is it true that for any $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert \geq \alpha N$ there exists some $S\subseteq A$ such that \[\frac{a}{b}=\sum_{n\in S}\frac{1}{n}\] with $a\leq b =O_\alpha(1)$?
Liu and Sawhney [LiSa24] observed that the main result of Bloom [Bl21] implies a positive solution to this conjecture. They prove a more precise version, that if $(\log N)^{-1/7+o(1)}\leq \alpha \leq 1/2$ then there is some $S\subseteq A$ such that \[\frac{a}{b}=\sum_{n\in S}\frac{1}{n}\] with $a\leq b \leq \exp(O(1/\alpha))$. They also observe that the dependence $b\leq \exp(O(1/\alpha))$ is sharp.
What is the minimal value of $\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$ as $A$ ranges over all subsets of $\{1,\ldots,N\}$ which contain no $S$ such that $\sum_{n\in S}\frac{1}{n}=1$? Is it \[e^{-(c+o(1))N}\] for some constant $c\in (0,1)$?
It is trivially at least $1/[1,\ldots,N]$.
Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of integers with $\sum_{n\in A}\frac{1}{n}>K$ there exists some $S\subseteq A$ such that \[1-e^{-cK} < \sum_{n\in S}\frac{1}{n}\leq 1?\]
Erdős and Graham knew this with $e^{-cK}$ replaced by $c/K^2$.
Additional thanks to: Mehtaab Sawhney
Are there infinitely many solutions to \[\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},\] where $m\geq 2$ is an integer and $p_1<\cdots<p_k$ are distinct primes?
For example, \[\frac{1}{2}+\frac{1}{3}=1-\frac{1}{6}.\]
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.

Additional thanks to: Quanyu Tang
Is it true that if $A\subset \mathbb{N}\backslash\{1\}$ is a finite set with $\sum_{n\in A}\frac{1}{n}<2$ then there is a partition $A=A_1\sqcup A_2$ such that \[\sum_{n\in A_i}\frac{1}{n}<1\] for $i=1,2$?
This is not true if $A$ is a multiset, for example $2,3,3,5,5,5,5$.

This is not true in general, as shown by Sándor [Sa97], who observed that the proper divisors of $120$ form a counterexample. More generally, Sándor shows that for any $n\geq 2$ there exists a finite set $A\subseteq \mathbb{N}\backslash\{1\}$ with $\sum_{k\in A}\frac{1}{k}<n$ and no partition into $n$ parts each of which has $\sum_{k\in A_i}\frac{1}{k}<1$.

The minimal counterexample is $\{2,3,4,5,6,7,10,11,13,14,15\}$, found by Tom Stobart.

Additional thanks to: Tom Stobart
Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with \[0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?\] Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$, \[\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}\] whenever the left-hand side is not zero?
Inequality is obvious for the second claim, the problem is strict inequality. This fails for small $n$, for example \[\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.\]
Additional thanks to: Zachary Chase
Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a finite $S\subset A$ such that \[\sum_{n\in S}\frac{f(n)}{n}=0?\] What about if $A$ is an arbitrary set of positive density? What if $A$ is the set of squares excluding $1$?
Erdős and Straus [ErSt75] proved this when $A=\mathbb{N}$. Sattler [Sat75] proved this when $A$ is the set of odd numbers. For the squares $1$ must be excluded or the result is trivially false, since \[\sum_{k\geq 2}\frac{1}{k^2}<2.\]
What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that \[\sum_{n\in A}\frac{\delta_n}{n}=0\] and \[\sum_{n\in A'}\frac{\delta_n}{n}\neq 0\] for all $A'\subsetneq A$?
Let $S(N)$ count the number of distinct sums of the form $\sum_{n\in A}\frac{1}{n}$ for $A\subseteq \{1,\ldots,N\}$. Estimate $S(N)$.
Bleicher and Erdős [BlEr75] proved the lower bound \[\frac{N}{\log N}\prod_{i=3}^k\log_iN\leq \frac{\log S(N)}{\log 2},\] valid for $k\geq 4$ and $\log_kN\geq k$, and also [BlEr76b] proved the upper bound \[\log S(N)\leq \log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right),\] valid for $r\geq 1$ and $\log_{2r}N\geq 1$. (In these bounds $\log_in$ denotes the $i$-fold iterated logarithm.)

See also [321].

Additional thanks to: Boris Alexeev and Dustin Mixon
What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that all sums $\sum_{n\in S}\frac{1}{n}$ are distinct for $S\subseteq A$?
Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that \[\frac{N}{\log N}\prod_{i=3}^k\log_iN\leq R(N)\leq \frac{1}{\log 2}\log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right),\] valid for any $k\geq 4$ with $\log_kN\geq k$ and any $r\geq 1$ with $\log_{2r}N\geq 1$. (In these bounds $\log_in$ denotes the $i$-fold iterated logarithm.)

See also [320].

Additional thanks to: Boris Alexeev, Zachary Hunter, and Dustin Mixon
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 $f_{k,k}(x)=o(x)$.

Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a<b$ nonnegative integers are distinct?
Erdős and Graham describe this problem as 'very annoying'. Probably $f(x)=x^5$ should work.
Let $k\geq 3$ and $f_{k,3}(x)$ denote the number of integers $\leq x$ which are the sum of three nonnegative $k$th powers. Is it true that \[f_{k,3}(x) \gg x^{3/k}\] or even $\gg_\epsilon x^{3/k-\epsilon}$?
Mahler and Erdős [ErMa38] proved that $f_{k,2}(x) \gg x^{2/k}$. For $k=3$ the best known is due to Wooley [Wo15], \[f_{3,3}(x) \gg x^{0.917\cdots}.\]
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 $C$.
Suppose $A\subseteq \mathbb{N}$ is a Sidon set. How large can \[\limsup_{N\to \infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}\] be?
Erdős proved that $1/2$ is possible and Krückeberg [Kr61] proved $1/\sqrt{2}$ is possible. Erdős and Turán [ErTu41] have proved this $\limsup$ is always $\leq 1$.

The fact that $1$ is possible would follow if any finite Sidon set is a subset of a perfect difference set.

Suppose $A\subset\mathbb{N}$ is a minimal basis with positive density. Is it true that, for any $n\in A$, the (upper) density of integers which cannot be represented without using $n$ is positive?
Asked by Erdős and Nathanson.
Let $A,B\subseteq \mathbb{N}$ such that for all large $N$ \[\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}\] and \[\lvert B\cap \{1,\ldots,N\}\rvert \gg N^{1/2}.\] Is it true that there are infinitely many solutions to $a_1-a_2=b_1-b_2\neq 0$ with $a_1,a_2\in A$ and $b_1,b_2\in B$?
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 $D(A)\neq\emptyset$.

Let $A\subseteq \mathbb{N}$ be a set of density zero. Does there exist a basis $B$ such that $A\subseteq B+B$ and \[\lvert B\cap \{1,\ldots,N\}\rvert =o(N^{1/2})\] for all large $N$?
Erdős and Newman [ErNe77] have proved this is true when $A$ is the set of squares.
Find the best function $f(n)$ such that every $n$ can be written as $n=a+b$ where both $a,b$ are $f(n)$-smooth (that is, are not divisible by any prime $p>f(n)$.)
Erdős originally asked if even $f(n)\leq n^{1/3}$ is true. This is known, and the best bound is due to Balog [Ba89] who proved that \[f(n) \ll_\epsilon n^{\frac{4}{9\sqrt{e}}+\epsilon}\] for all $\epsilon>0$. (Note $\frac{4}{9\sqrt{e}}=0.2695\ldots$.)

It is likely that $f(n)\leq n^{o(1)}$, or even $f(n)\leq e^{O(\sqrt{\log n})}$.

Additional thanks to: Zachary Hunter
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 $\geq 7$).
Additional thanks to: Zachary Hunter
Let $A\subseteq \mathbb{N}$ be an additive basis (of any finite order) such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that \[\lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}=\infty?\]
The answer is no, and a counterexample was provided by Turjányi [Tu84]. This was generalised (to the replacement of $A+A$ by the $h$-fold sumset $hA$ for any $h\geq 2$) by Ruzsa and Turjányi [RT85]. Ruzsa and Turjányi do prove (under the same hypotheses) that \[\lim_{N\to \infty}\frac{\lvert (A+A+A)\cap \{1,\ldots,3N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}=\infty,\] and conjecture that the same should be true with $(A+A)\cap \{1,\ldots,2N\}$ in the numerator.
Additional thanks to: Zachary Chase, Zachary Hunt
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 $A\subseteq \mathbb{N}$ be a basis of order $r$. Must the set of integers representable as the sum of exactly $r$ distinct elements from $A$ have positive density?
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.

Additional thanks to: Zachary Hunter
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 $B$.
Is it true that if $a_{n+1}/a_n \geq 1+\epsilon$ for some $\epsilon>0$ and all $n$ then \[\lim_n \frac{a_{n+1}}{a_n}=\frac{1+\sqrt{5}}{2}?\]
Graham [Gr64d] has shown that the sequence $a_n=F_n-(-1)^{n}$, where $F_n$ is the $n$th Fibonacci number, has these properties. Erdős and Graham [ErGr80] remark that it is easy to see that if $a_{n+1}/a_n>\frac{1+\sqrt{5}}{2}$ then the second property is automatically satisfied, and that it is not hard to construct very irregular sequences satisfying both properties.
Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with \[\lim \frac{a_{n+1}}{a_n}=2\] such that \[P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}\] has density $1$ for every cofinite subsequence $A'$ of $A$?
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.
Additional thanks to: Vjekoslav Kovac
Let $A\subseteq \mathbb{R}^2$ be a measurable set with infinite measure. Must $A$ contain the vertices of an isosceles trapezoid of area $1$?
Erdős and Mauldin (unpublished) claim that this is true for trapezoids in general, but fails for parallelograms.
Additional thanks to: Vjekoslav Kovac
Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is \[\{ \lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 4\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 4\beta\rfloor,\ldots\}\] complete? What if $2$ is replaced by some $\gamma\in(1,2)$?
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 $n$)?

Additional thanks to: Adrian Beker
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 $a_1<a_2<\cdots$ be an infinite sequence of integers such that $a_1=k$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. What can be said about the density of this sequence?
A problem of MacMahon, studied by Andrews [An75].
Let $f(n)$ be minimal such that $\{1,\ldots,n\}$ can be partitioned into $f(n)$ classes so that $n$ cannot be expressed as a sum of distinct elements from the same class. How fast does $f(n)$ grow?
Erdős and Graham state it 'can be shown' that $f(n)\to \infty$.
Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?
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 $t=\tfrac{1}{2}\sum_{n\in A}n$.
Additional thanks to: Zachary Chase
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 $x^2=8y^2+1$.
Additional thanks to: Zachary Chase
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 $\epsilon>0$, \[\limsup \frac{\prod_{n\leq m<n+k}B_r(m) }{n^{1+\epsilon}}\to\infty?\]
How large is the largest prime factor of $n(n+1)$?
Mahler [Ma35] showed that this is $\gg \log\log n$. Schinzel [Sc67b] observed that for infinitely many $n$ it is $\leq n^{O(1/\log\log\log n)}$. The truth is probably $\gg (\log n)^2$ for all $n$.
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.
Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n)>P(n+1)$ has density $1/2$.
Conjectured by Erdős and Pomerance [ErPo78], who proved that this set and its complement both have positive upper density.
Let $P(n)$ denote the largest prime factor of $n$. There are infinitely many $n$ such that $P(n)>P(n+1)>P(n+2)$.
Conjectured by Erdős and Pomerance [ErPo78], who proved the analogous result for $P(n)<P(n+1)<P(n+2)$. Proved by Balog [Ba01], who proved that this is true for $\gg \sqrt{x}$ many $n\leq x$ (for all large $x$).
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 $k>6$.
Additional thanks to: Zachary Chase
Let $n+1,\ldots,n+k$ be consecutive composite integers. Are there distinct primes $p_1,\ldots,p_k$ such that $p_i\mid n+i$ for $1\leq i\leq k$?
Very deep if true, since it would imply there is a prime between any two consecutive squares.

Originally conjectured by Grimm [Gr69], who proved it when $k \ll \log n/\log\log n$. Ramachandra, Shorey, and Tijdeman [RST75] have improved this to \[k \ll \left(\frac{\log n}{\log\log n}\right)^3.\]

Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$?
Erdős, Graham, Ruzsa, and Straus [EGRS75] have shown that, for any two primes $p$ and $q$, there are infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $pq$.
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].

The product of more than two consecutive integers is never powerful (a number $n$ is powerful if whenever $p\mid n$ we have $p^2\mid n$).
Conjectured by Erdős and Selfridge. There are infinitely many $n$ such that $n(n+1)$ is powerful (see [364]).
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].

Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of \[\prod_{0\leq i\leq k}(p^2+i)\] is $p$?
If $1<k<n-1$ then $\binom{n}{k}$ is divisible by a prime $p<n/2$ (except $\binom{7}{3}=5\cdot 7$).
A conjecture of Erdős and Selfridge. Proved by Ecklund [Ec69], who made the stronger conjecture that whenever $n>k^2$ the binomial coefficient $\binom{n}{k}$ is divisible by a prime $p<n/k$. They have proved the weaker inequality $p\ll n/k^c$ for some constant $c>0$.
Additional thanks to: Zachary Chase
Let \[F(n) = \max_{\substack{m<n\\ m\textrm{ composite}}} m+p(m),\] where $p(m)$ is the least prime divisor of $m$. Is it true that $F(n)>n$ for all sufficiently large $n$? Does $F(n)-n\to \infty$ as $n\to\infty$?
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].
Additional thanks to: Zachary Chase
Can one classify all solutions of \[\prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j)\] where $1<k_1<k_2$ and $m_1+k_1\leq m_2$? Are there only finitely many solutions? More generally, if $k_1>2$ then for fixed $a$ and $b$ \[a\prod_{1\leq i\leq k_1}(m_1+i)=b\prod_{1\leq j\leq k_2}(m_2+j)\] should have only a finite number of solutions. What if one just requires that the products have the same prime factors, say when $k_1=k_2$?
Is it true that for every $n$ there is a $k$ such that \[\prod_{1\leq i\leq k}(n+i) \mid \prod_{1\leq i\leq k}(n+k+i)?\]
Asked by Erdős and Straus. For example when $n=2$ we have $k=4$: \[3\cdot 4\cdot 5\cdot 6 \mid 7\cdot 8\cdot 9\cdot 10.\] No example was known for $n\geq 10$.
Let $f(n)$ be the minimal $m$ such that \[n! = a_1\cdots a_k\] with $n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant $c$ such that \[f(n)-2n \sim c\frac{n}{\log n}?\]
Erdős, Guy, and Selfridge [EGS82] have shown that \[f(n)-2n \asymp \frac{n}{\log n}.\]
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 $k\geq 3$.
Additional thanks to: Zachary Chase
Prove that there are no $n,x,y$ with $x+n\leq y$ such that \[\mathrm{lcm}(x+1,\ldots,x+n) = \mathrm{lcm}(y+1,\ldots,y+n).\]
It follows from the Thue-Siegel theorem that, for $n$ fixed, there are only finitely many potential such $x$ and $y$.
Is it true that for every $k$ there exists $n$ such that \[\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?\]
Erdős and Graham write that $n+1$ always divides $\binom{2n}{n}$, but it is quite rare that $n$ divides $\binom{2n}{n}$.
Are there only finitely many solutions to \[\prod_i \binom{2m_i}{m_i}=\prod_j \binom{2n_j}{n_j}\] with the $m_i,n_j$ distinct?
Are the only solutions to \[n!=x^2-1\] when $n=4,5,7$?
The Brocard-Ramanujan conjecture. Erdős and Graham describe this as an old conjecture, and write it 'is almost certainly true but it is intractable at present'.

Overholt [Ov93] has shown that this has only finitely many solutions assuming a weak form of the abc conjecture.

Is it true that there are no solutions to \[n! = x^k\pm y^k\] with $x,y,n\in \mathbb{N}$, with $xy>1$ and $k>2$?
Erdős and Obláth [ErOb37] proved this is true when $(x,y)=1$ and $k\neq 4$. Pollack and Shapiro [PoSh73] proved this is true when $(x,y)=1$ and $k=4$. The known methods break down without the condition $(x,y)=1$.
Additional thanks to: Zachary Chase
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].

Is there some function $\omega(r)$ such that $\omega(r)\to \infty$ as $r\to\infty$, such that for all large $n$ there exist $a_1,a_2$ with \[a_1+a_2> n+\omega(r)\log n\] such that $a_1!a_2! \mid n!2^n3^n\cdots p_r^n$?
See also [400].
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].

Let $f(a,p)$ be the largest $k$ such that there are $a=a_1<\cdots<a_k$ such that \[p^k \mid (a_1!+\cdots+a_k!).\] Is $f(a,p)$ bounded by some absolute constant? What if this constant is allowed to depend on $a$ and $p?

Is there a prime $p$ and an infinite sequence $a_1<a_2<\cdots$ such that if $p^{m_k}$ is the highest power of $p$ dividing $\sum_{i\leq k}a_i!$ then $m_k\to \infty$?

See also [403]. Lin [Li76] has shown that $f(2,2) \leq 254$.
Let $p$ be a prime. Is it true that the equation \[(p-1)!+a^{p-1}=p^k\] has only finitely many solutions?
Erdős and Graham remark that it is probably true that in general $(p-1)!+a^{p-1}$ is rarely a power at all (although this can happen, for example $6!+2^6=28^2$).
Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?
The only examples seem to be $4=1+3$ and $256=1+3+3^2+3^5$. If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power of $2$.
Let $w(n)$ count the number of solutions to \[n=2^a+3^b+2^c3^d\] with $a,b,c,d\geq 0$ integers. Is it true that $w(n)$ is bounded by some absolute constant?
A conjecture originally due to Newman.

This is true, and was proved by Evertse, Györy, Stewart, and Tijdeman [EGST88].

Let $\phi(n)$ be the Euler totient function and $\phi_k(n)$ be the iterated $\phi$ function, so that $\phi_1(n)=\phi(n)$ and $\phi_k(n)=\phi(\phi_{k-1}(n))$. Let \[f(n) = \min \{ k : \phi_k(n)=1\}.\] Does $f(n)/\log n$ have a distribution function? Is $f(n)/\log n$ almost always constant? What can be said about the largest prime factor of $\phi_k(n)$ when, say, $k=\log\log n$?
Pillai [Pi29] was the first to investigate this function, and proved \[\log_3 n < f(n) < \log_2 n\] for all large $n$. Shapiro [Sh50] proved that $f(n)$ is essentially multiplicative.

Erdős, Granville, Pomerance, and Spiro [EGPS90] have proved that the answer to the first two questions is yes, conditional on a form of the Elliott-Halberstam conjecture.

It is likely true that, if $k\to \infty$ however slowly with $n$, then for almost $n$ the largest prime factor of $\phi_k(n)$ is $\leq n^{o(1)}$.

Additional thanks to: Zachary Chase
How many iterations of $n\mapsto \phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density of $n$ which reach any fixed prime?
A problem of Finucane. One can also ask about $n\mapsto \sigma(n)-1$.
Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that \[\lim_{k\to \infty} \sigma_k(n)^{1/k}=\infty?\]
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'.

See also [413] and [414].

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 $m<n$?

Let $h_1(n)=h(n)=n+\tau(n)$ (where $\tau(n)$ counts the number of divisors of $n$) and $h_k(n)=h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist $i$ and $j$ such that $h_i(m)=h_j(n)$?
Asked by Spiro. That is, there is (eventually) only one possible sequence that the iterations of $n\mapsto h(n)$ can settle on. Erdős and Graham believed the answer is yes. Similar questions can be asked by the iterates of many other functions. See also [412] and [413].
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].

Let $V'(x)=\#\{\phi(m) : m\leq x\}$, and $V(x)$ count the number of $n\leq x$ such that $\phi(m)=n$ is solvable. Does $\lim V(x)/V'(x)$ exist? Is it $>1$?
It is trivial that $V'(x) \leq V(x)$. See also [416].
Are there infinitely many integers not of the form $n-\phi(n)$?
Asked by Erdős and Sierpiński. It follows from the Goldbach conjecture that every odd number can be written as $n-\phi(n)$. What happens for even numbers?

Erdős [Er73b] has shown that a positive density set of integers cannot be written as $\sigma(n)-n$.

This is true, as shown by Browkin and Schinzel [BrSc95], who show that any integer of the shape $2^{k}\cdot 509203$ is not of this form. It seems to be open whether there is a positive density set of integers not of this form.

Additional thanks to: Stefan Steinerberger
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.
Additional thanks to: Zachary Chase
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 $c>0$.
Is there a sequence $1\leq d_1<d_2<\cdots$ with density $1$ such that all products $\prod_{u\leq i\leq v}d_i$ are distinct?
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 $A\subseteq \{1,\ldots,N\}$ of size $\ll N/\log N$ such that every $m\leq N$ can be written as $a+2^k$ for some $k\geq 0$ and $a\in A$?
The answer is yes, proved by Ruzsa [Ru72].
Is it true that, for every $n$ and $d$, there exists $k$ such that \[d \mid p_{n+1}+\cdots+p_{n+k},\] where $p_r$ denotes the $r$th prime?
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 $\limsup$.
Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists some $n$ such that $n+a$ is prime for all $a\in A$?
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].

Let $A,B\subseteq \mathbb{N}$ be two infinite sets. How dense can $A+B$ be if all elements of $A+B$ are pairwise relatively prime?
Asked by Straus, inspired by a problem of Ostmann (see [431]).
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 $t\to \infty$.

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

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

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

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

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

Let $G$ be the infinite graph on $\mathbb{N}$ where we connect $m,n$ by an edge if and only if $n+m$ is a square. Is the chromatic number of $G$ equal to $\aleph_0$? What if $n+m$ is required to be a $k$th power?
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].
Additional thanks to: Zachary Chase
Let $N\geq 1$. What is the size of the largest $A\subset \{1,\ldots,N\}$ such that $\mathrm{lcm}(a,b)\leq N$ for all $a,b\in A$? Is it attained by choosing all integers in $[1,(N/2)^{1/2}]$ together with all even integers in $[(N/2)^{1/2},(2N)^{1/2}]$?
Is it true that if $A\subseteq\mathbb{N}$ is such that \[\frac{1}{\log\log x}\sum_{n\in A\cap [1,x)}\frac{1}{n}\to \infty\] then \[\left(\sum_{n\in A\cap [1,x)}\frac{1}{n}\right)^{-2} \sum_{\substack{a,b\in A\cap (1,x]\\ a<b}}\frac{1}{\mathrm{lcm}(a,b)}\to \infty?\]
Let $m,n\geq 1$. What is \[\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}?\] Can it be arbitrarily large? Is it $\leq (mn)^{o(1)}$ for all suffiicently large $m,n$?
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].
Is it true that, for any $c>1/2$, if $p$ is a large prime and $n$ is sufficiently large (both depending on $c$) then there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$?
An unpublished result of Heilbronn guarantees this for $c$ sufficiently close to $1$.
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.

Additional thanks to: Zachary Chase
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 $r(n)$ count the number of $d_1,d_2$ such that $d_1\mid n$ and $d_2\mid n$ and $d_1<d_2<2d_1$. Is it true that, for every $\epsilon>0$, \[r(n) < \epsilon \tau(n)\] for almost all $n$, where $\tau(n)$ is the number of divisors of $n$?
See also [447].
How large must $y=y(\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\epsilon y$?
Estimate $n_k$, the smallest integer such that $\prod_{1\leq i\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.
Erdős and Graham write 'we can prove $n_k>k^{1+c}$ but no doubt much more is true'.
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 $k$.
Is it true that, for all sufficiently large $n$, \[p_n^2 \leq p_{n+i}p_{n-i}\] for all $i<n$, where $p_k$ is the $k$th prime?
Asked by Erdős and Straus. The answer is no, as shown by Pomerance [Po79].
Let \[f(n) = \min_{i<n} (p_{n+i}+p_{n-i}),\] where $p_k$ is the $k$th prime. Is it true that \[\limsup_n (f(n)-2p_n)=\infty?\]
Pomerance [Po79] has proved the $\limsup$ is at least $2$.
Let $q_1<q_2<\cdots$ be a sequence of primes such that \[q_{n+1}-q_n\geq q_n-q_{n-1}.\] Must \[\lim_n \frac{q_n}{n^2}=\infty?\]
Richter [Ri76] proved that \[\liminf_n \frac{q_n}{n^2}>2.84\cdots.\]
Let $s_n$ be the smallest prime such that $n\mid s_n-1$ and let $m_n$ be the smallest integer such that $n\mid \phi(m_n)$. Is it true that $s_n>m_n$ for almost all $n$? Does $s_n/m_n\to \infty$ for almost all $n$? Are there infinitely many primes $p$ such that $p-1$ is the only $n$ for which $m_n=p$?
Is there some $\epsilon>0$ such that there are infinitely many $n$ where all primes $p\leq (2+\epsilon)\log n$ divide \[\prod_{1\leq i\leq \log n}(n+i)?\]
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.
Additional thanks to: Zachary Chase
Let $f(u)$ be the largest $v$ such that no $m\in (u,v)$ is composed entirely of primes dividing $uv$. Estimate $f(u)$.
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 $x$?
Additional thanks to: Zachary Chase
Is there a function $f$ with $f(n)\to \infty$ as $n\to \infty$ such that, for all large $n$, there is a composite number $m$ such that \[n+f(n)<m<n+p(m)?\] (Here $p(m)$ is the least prime factor of $m$.)
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.
Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that \[\| \lvert P_i-P_j\rvert \| \geq \delta\] for all $1\leq i<j\leq n$. (Here $\|x\|$ is the distance from $x$ to the nearest integer.)

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

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

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

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

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

Graham proved this is true, and in fact \[N(X,1/10)> \frac{\log X}{10}.\] This was substantially improved by Sárközy [Sa76], who proved that for, all sufficiently small $\delta>0$, \[N(X,\delta)>X^{1/2-\delta^{1/7}}.\] See also [465].
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.
For any $n$ let $D_n$ be the set of sums of the shape $d_1,d_1+d_2,d_1+d_2+d_3,\ldots$ where $1<d_1<d_2<\cdots$ are the divisors of $n$.

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

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

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.
Additional thanks to: Zachary Chase
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.
Additional thanks to: Zachary Chase
Let $A\subseteq \mathbb{F}_p$. Let \[A\hat{+}A = \{ a+b : a\neq b \in A\}.\] Is it true that \[\lvert A\hat{+}A\rvert \geq \min(2\lvert A\rvert-3,p)?\]
A question of Erdős and Heilbronn. Solved in the affirmative by da Silva and Hamidoune [dSHa94].
Let $f:\mathbb{Z}\to \mathbb{Z}$ be a polynomial of degree at least $2$. Is there a set $A$ such that every $z\in \mathbb{Z}$ has exactly one representation as $z=a+f(n)$ for some $a\in A$ and $n\in \mathbb{Z}$?
Probably there is no such $A$ for any polynomial $f$.
Let $p$ be a prime and \[A_p = \{ k! \mod{p} : 1\leq k<p\}.\] Is it true that \[\lvert A_p\rvert \sim (1-\tfrac{1}{e})p?\]
Additional thanks to: Zachary Chase
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'.
Define a sequence by $a_1=1$ and \[a_{n+1}=\lfloor\sqrt{2}(a_n+1/2)\rfloor\] for $n\geq 1$. The difference $a_{2n+1}-2a_{2n-1}$ is the $n$th digit in the binary expansion of $\sqrt{2}$.

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

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

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

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

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

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

Let $A$ be a finite set and \[B=\{ n \geq 1 : a\nmid n\textrm{ for all }a\in A\}.\] Is it true that, for every $m>n$, \[\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?\]
The example $A=\{a\}$ and $n=2a-1$ and $m=2a$ shows that $2$ would be best possible.
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].

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

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

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

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

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

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

Does there exist a $k$ such that every sufficiently large integer can be written in the form \[\prod_{i=1}^k a_i - \sum_{i=1}^k a_i\] for some integers $a_i\geq 2$?
A conjecture of Schinzel.
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 $l\geq 0$).
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].

Additional thanks to: Zachary Chase
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 $M=(a_{ij})$ be a real $n\times n$ doubly stochastic matrix (i.e. the entries are non-negative and each column and row sums to $1$). Does there exist some $\sigma\in S_n$ such that \[\prod_{1\leq i\leq n}a_{i\sigma(i)}\geq n^{-n}?\]
A weaker form of the conjecture of van der Waerden, which states that \[\mathrm{perm}(M)=\sum_{\sigma\in S_n}\prod_{1\leq i\leq n}a_{i\sigma(i)}\geq n^{-n}n!\] with equality if and only if $a_{ij}=1/n$ for all $i,j$.

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

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

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

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 $\leq \exp(n^{1-o(1)})$.
What is the size of the largest $A\subseteq \mathbb{R}^n$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $x,y,z$ from $A$, at least two of the distances $\lvert x-y\rvert,\lvert y-z\rvert,\lvert x-z\rvert$ are equal.
When $n=2$ the answer is $6$ (due to Kelly [ErKe47]). When $n=3$ the answer is $8$ (due to Croft [Cr62]). The best upper bound known in general is due to Blokhuis [Bl84] who showed that \[\lvert A\rvert \leq \binom{n+2}{2}.\]
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 $c>1$.

What is the minimum number of circles determined by any $n$ points in $\mathbb{R}^2$, not all on a circle?
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 $f(z)\in\mathbb{C}[z]$ be a monic polynomial. Can the set \[\{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\}\] be covered by a set of circles the sum of whose radii is $\leq 2$?
Cartan proved this is true with $2$ replaced by $2e$, which was improved to $2.59$ by Pommerenke [Po61]. Pommerenke [Po59] proved that $2$ is achievable if the set is connected (in fact the entire set is covered by a single circle with radius $2$).
If $A\subset \mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that \[\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?\]
Chowla's cosine problem. The best known bound currently, due to Ruzsa [Ru04] (improving on an earlier result of Bourgain [Bo86]), replaces $N^{1/2}$ by \[\exp(O(\sqrt{\log N}).\] The example $A=B-B$, where $B$ is a Sidon set, shows that $N^{1/2}$ would be the best possible here.
Let $f(z)\in \mathbb{C}[z]$ be a monic polynomial of degree $n$ and \[A = \{ z\in \mathbb{C} : \lvert f(z)\rvert\leq 1\}.\] Is it true that, for every such $f$ and constant $c>0$, the set $A$ can have at most $O_c(1)$ many components of diameter $>1+c$ (where the implied constant is in particular independent of $n$)?
Is it true that, if $A\subset \mathbb{Z}$ is a finite set of size $N$, then \[\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N,\] where $e(x)=e^{2\pi ix }$?
Littlewood's conjecture, proved independently by Konyagin [Ko81] and McGehee, Pigno, and Smith [MPS81].
Let $f=\sum_{n=0}^\infty a_nz^n$ be an entire function. What is the greatest possible value of \[\liminf_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}?\]
It is trivial that this value is in $[1/2,1)$. Kövári (unpublished) observed that it must be $>1/2$. Clunie and Hayman [ClHa64] showed that it is $\leq 2/\pi-c$ for some absolute constant $c>0$. Some other results on this quantity were established by Gray and Shah \cit{GrSh63}.

See also [227].

Let $f(z)$ be an entire function. Does there exist a path $L$ so that, for every $n$, \[\lvert f(z)/z^n\rvert \to \infty\] as $z\to \infty$ along $L$?

Can the length of this path be estimated in terms of $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$? Does there exist a path along which $\lvert f(z)\rvert$ tends to $\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\epsilon$)?

Boas (unpublished) has proved the first part, that such a path must exist.
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 $f(z)=\sum_{k\geq 1}a_k z^{n_k}$ be an entire function of finite order such that $\lim n_k/k=\infty$. Let $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$ and $m(r)=\max_n \lvert a_nr^n\rvert$. Is it true that \[\limsup\frac{\log m(r)}{\log M(r)}=1?\]
A problem of Pólya. Results of Wiman [Wi14] imply that if $(n_{k+1}-n_k)^2>n_k$ then $\limsup \frac{m(r)}{M(r)}=1$. Erdős and Macintyre [ErMa54] proved this under the assumption that \[\sum_{k\geq 2}\frac{1}{n_{k+1}-n_k}<\infty.\]
Let $f(z)=\sum_{k=1}^\infty a_kz^{n_k}$ be an entire function. Is it true that if $n_k/k\to \infty$ then $f(z)$ assumes every value infinitely often?
A conjecture of Fejér and Pólya. Fejér [Fe08] proved that if $\sum\frac{1}{n_k}<\infty$ then $f(z)$ assumes every value at least once, and Biernacki [Bi28] showed that this holds under the assumption that $n_k/k\to \infty$.
Let $A_1,A_2,\ldots$ be sets of complex numbers, none of which has a limit point in $\mathbb{C}$. Does there exist an entire function $f(z)$ and a sequence $n_1<n_2<\cdots$ such that $f^{(n_k)}$ vanishes on $A_k$ for all $k\geq 1$?
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 Rademacher multiplicative function: a random $\{-1,0,1\}$-valued multiplicative function, where for each prime $p$ we independently choose $f(p)\in \{-1,1\}$ uniformly at random, and for square-free integers $n$ we extend $f(p_1\cdots p_r)=f(p_1)\cdots f(p_r)$ (and $f(n)=0$ if $n$ is not squarefree). Does there exist some constant $c>0$ such that, almost surely, \[\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N\log\log N}}=c?\]
Note that if we drop the multiplicative assumption, and simply assign $f(m)=\pm 1$ at random, then this statement is true (with $c=\sqrt{2}$), the law of the iterated logarithm.

Wintner [Wi44] proved that, almost surely, \[\sum_{m\leq N}f(m)\ll N^{1/2+o(1)},\] and Erdős improved the right-hand side to $N^{1/2}(\log N)^{O(1)}$. Lau, Tenenbaum, and Wu [LTW13] have shown that, almost surely, \[\sum_{m\leq N}f(m)\ll N^{1/2}(\log\log N)^{2+o(1)}.\] Harper [Ha13] has shown that the sum is almost surely not $O(N^{1/2}/(\log\log N)^{5/2+o(1)})$, and conjectured that in fact Erdős' conjecture is false, and almost surely \[\sum_{m\leq N}f(m) \ll N^{1/2}(\log\log N)^{1/4+o(1)}.\] This was proved by Caich [Ca23].

Additional thanks to: Mehtaab Sawhney
For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). Let $R_n(t)$ denote the number of real roots of $\sum_{1\leq k\leq n}\epsilon_k(t)z^k$. Is it true that, for almost all $t\in (0,1)$, we have \[\lim_{n\to \infty}\frac{R_n(t)}{\log n}=\frac{\pi}{2}?\]
Erdős and Offord [EO56] showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.

See also [522].

Is it true that all except $o(2^n)$ many polynomials of degree $n$ with $\pm 1$-valued coefficients have $(\frac{1}{2}+o(1))n$ many roots in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$?

For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). If $S_n(t)$ is the number of roots of $\sum_{1\leq k\leq n}\epsilon_k(t)z^k$ in $\lvert z\rvert \leq1$ then is it true that, for almost all $t\in (0,1)$, \[\lim_{n\to \infty}\frac{S_n(t)}{n}=\frac{1}{2}?\]

Erdős and Offord [EO56] showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.

See also [521].

Additional thanks to: Zachary Chase
For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). Does there exist some constant $C>0$ such that, for almost all $t\in (0,1)$, \[\max_{\lvert z\rvert=1}\left\lvert \sum_{k\leq n}\epsilon_k(t)z^k\right\rvert=(C+o(1))\sqrt{n\log n}?\]
Salem and Zygmund [SZ54] proved that $\sqrt{n\log n}$ is the right order of magnitude, but not an asymptotic.
For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for almost all $t\in(0,1)$) for \[M_n(t)=\max_{x\in [0,1]}\left\lvert \sum_{k\leq n}\epsilon_k(t)x^k\right\rvert?\]
A problem of Salem and Zygmund [SZ54]. Chung showed that, for almost all $t$, there exist infinitely many $n$ such that \[M_n(t) \ll \left(\frac{n}{\log\log n}\right)^{1/2}.\] Erdős (unpublished) showed that for almost all $t$ and every $\epsilon>0$ we have $\lim_{n\to \infty}M_n(t)/n^{1/2-\epsilon}=\infty$.
Is it true that all except at most $o(2^n)$ many degree $n$ polynomials with $\pm 1$-valued coefficients $f(z)$ have $\lvert f(z)\rvert <1$ for some $\lvert z\rvert=1$? What is the behaviour of \[\min_{\lvert z\rvert=1}\lvert f(z)\rvert?\]
Let $a_n\geq 0$ with $a_n\to 0$ and $\sum a_n=\infty$. Find a necessary and sufficient condition on the $a_n$ such that, if we choose (independently and uniformly) random arcs on the unit circle of length $a_n$, then all the circle is covered with probability $1$.
A problem of Dvoretzky [Dv56]. It is easy to see that (under the given conditions alone) almost all the circle is covered with probability $1$.

Kahane [Ka59] showed that $a_n=\frac{1+c}{n}$ with $c>0$ has this property, which Erd\H{s} (unpublished) improved to $a_n=\frac{1}{n}$. Erd\{o}s also showed that $a_n=\frac{1-c}{n}$ with $c>0$ does not have this property.

Solved by Shepp [Sh72], who showed that a necessary and sufficient condition is that \[\sum_n \frac{e^{a_1+\cdots+a_n}}{n^2}=\infty.\]

Let $a_n\in \mathbb{R}$ be such that $\sum_n \lvert a_n\rvert^2=\infty$ and $\lvert a_n\rvert=o(1/\sqrt{n})$. Is it true that, for almost all $\epsilon_n=\pm 1$, there exists some $z$ with $\lvert z\rvert=1$ (depending on the choice of signs) such that \[\sum_n \epsilon_n a_n z^n\] converges?
It is unclear to me whether Erdős also intended to assume that $\lvert a_{n+1}\rvert\leq \lvert a_n\rvert$.

It is 'well known' that, for almost all $\epsilon_n=\pm 1$, the series diverges for almost all $\lvert z\rvert=1$ (assuming only $\sum \lvert a_n\rvert^2=\infty$).

Dvoretzky and Erdős [DE59] showed that if $\lvert a_n\rvert >c/\sqrt{n}$ then, for almost all $\epsilon_n=\pm 1$, the series diverges for all $\lvert z\rvert=1$.

Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine \[C_k=\lim_{n\to\infty}f(n,k)^{1/n}.\]
The constant $C_k$ is sometimes known as the connective constant. Hammersley and Morton [HM54] showed that this limit exists, and it is trivial that $k\leq C_k\leq 2k-1$.

Kesten [Ke63] proved that $C_k=2k-1-1/2k+O(1/k^2)$, and more precise asymptotics are given by Clisby, Liang, and Slade [CLS07].

Conway and Guttmann [CG93] showed that $C_2\geq 2.62$ and Alm [Al93] showed that $C_2\leq 2.696$.

Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\mathbb{Z}^k$ (conditional on no self intersections). Is it true that \[\lim_{n\to \infty}\frac{d_2(n)}{n^{1/2}}= \infty?\] Is it true that \[d_k(n)\ll n^{1/2}\] for $k\geq 3$?
Let $\ell(N)$ be maximal such that in any finite set $A\subset \mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\ell(N)$ (i.e. the only solutions to $a+b=c+d$ in $S$ are the trivial ones). Determine the order of $\ell(N)$.
Originally asked by Riddell [Ri69]. Erdős noted the bounds \[N^{1/3} \ll \ell(N) \leq (1+o(1))N^{1/2}\] (the upper bound following from the case $A=\{1,\ldots,N\}$). The lower bound was improved to $N^{1/2}\ll \ell(N)$ by Komlós, Sulyok, and Szemerédi [KSS75]. The correct constant is unknown, but it is likely that the upper bound is true, so that $\ell(N)\sim N^{1/2}$.
Let $F(k)$ be the minimal $N$ such that if we two-colour $\{1,\ldots,N\}$ there is a set $A$ of size $k$ such that all subset sums $\sum_{a\in S}a$ (for $\emptyset\neq S\subseteq A$) are monochromatic. Estimate $F(k)$.
The existence of $F(k)$ was established by Sanders and Folkman, and it also follows from Rado's theorem. It is commonly known as Folkman's theorem.

Erdős and Spencer [ErSp89] proved that \[F(k) \geq 2^{ck^2/\log k}\] for some constant $c>0$. Balogh, Eberhrad, Narayanan, Treglown, and Wagner [BENTW17] have improved this to \[F(k) \geq 2^{2^{k-1}/k}.\]

If $\mathbb{N}$ is 2-coloured then is there some infinite set $A\subseteq \mathbb{N}$ such that all finite subset sums \[ \sum_{n\in S}n\] (as $S$ ranges over all non-empty finite subsets of $A$) are monochromatic?
In other words, must some colour class be an IP set. Asked by Graham and Rothschild. See also [531].

Proved by Hindman [Hi74] (for any number of colours).

Prove that if $G$ is a graph on $n$ vertices such that every set of $\lfloor n/2\rfloor$ vertices contains more than $n^2/50$ edges, then $G$ contains a triangle.
A problem of Erdős and Rousseau. The constant $50$ would be best possible as witnessed by a blow-up of $C_5$ or the Petersen graph. Krivelevich [Kr95] has proved this with $n/2$ replaced by $3n/5$ (and $50$ replaced by $25$).

Keevash and Sudakov [KeSu06] have proved this under the additional assumption that $G$ has at most $n^2/12$ edges.

See also the entry in the graphs problem collection.

What is the largest possible subset $A\subseteq\{1,\ldots,N\}$ which contains $N$ such that $(a,b)=1$ for all $a\neq b\in A$?
A problem of Erdős and Graham. They conjecture that this maximum is either $N/p$ (where $p$ is the smallest prime factor of $N$) or it is the number of integers $\{2t: t\leq N/2\textrm{ and }(t,N)=1\}$.
Let $r\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Estimate $f_r(N)$.
Erdős [Er64] proved that \[f_r(N) \leq N^{\frac{3}{4}+o(1)},\] and Abbott and Hanson [AbHa70] improved this exponent to $1/2$. Erdős [Er64] proved the lower bound \[f_3(N) > N^{\frac{c}{\log\log N}}\] for some constant $c>0$, and conjectured this should also be an upper bound.

Erdős writes this is 'intimately connected' with the sunflower problem [20]. Indeed, the conjectured upper bound would follow from the following stronger version of the sunflower problem: estimate the size of the largest $A$ such that if $\omega(n)=k$ for all $n\in A$ then there must be some $r$ many $a_1,\ldots,a_r\in A$ and an integer $d$ such that $(a_i,a_j)=d$ for all $i\neq j$ and $(a_i/d,d)=1$ for all $i$. The conjectured upper bound for $f_r(N)$ would follow if the size of such an $A$ must be at most $c_r^k$. The original sunflower proof of Erdős and Rado gives the upper bound $c_r^kk!$.

Let $\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\subseteq \{1,\ldots,N\}$ has size at least $\epsilon N$ then there must be $a,b,c\in A$ such that \[[a,b]=[b,c]=[a,c],\] where $[a,b]$ denotes the least common multiple?
This is false if we ask for four elements with the same pairwise least common multiple, as shown by Erdős [Er70].
Let $\epsilon>0$ and $N$ be sufficiently large. If $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert \geq \epsilon N$ then must there exist $a_1,a_2,a_3\in A$ and distinct primes $p_1,p_2,p_3$ such that \[a_1p_1=a_2p_2=a_3p_3?\]
A positive answer would imply [536].

Unfortunately Ruzsa disproved this, noting that the set $A$ of all squarefree numbers of the shape $p_1\cdots p_r$ where $p_{i+1}>2p_i$ for $1\leq i<r$ has positive density and disproves the conjecture, since for any $m$ if $m=pa$ where $p$ is prime and $a\in A$ then there are most 2 possibilities for $p$.

Let $r\geq 2$ and suppose that $A\subseteq\{1,\ldots,N\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\in A$. Give the best possible upper bound for \[\sum_{n\in A}\frac{1}{n}.\]
Erdős observed that \[\sum_{n\in A}\frac{1}{n}\sum_{p\leq N}\frac{1}{p}\leq r\sum_{m\leq N^2}\frac{1}{m}\ll r\log N,\] and hence \[\sum_{n\in A}\frac{1}{n} \ll r\frac{\log N}{\log\log N}.\] See also [536] and [537].
Let $h(n)$ be such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set \[\left\{ \frac{a}{(a,b)}: a,b\in A\right\}\] has size at least $h(n)$. Estimate $h(n)$.
Erdős and Szemerédi proved that \[n^{1/2} \ll h(n) \ll n^{1-c}\] for some constant $c>0$.
Is it true that if $A\subseteq \mathbb{Z}/N\mathbb{Z}$ has size $\gg N^{1/2}$ then there exists some non-empty $S\subseteq A$ such that $\sum_{n\in S}n\equiv 0\pmod{N}$?
The answer is yes, proved by Szemerédi [Sz70] (in fact for arbitrary finite abelian groups). Erdős speculates that perhaps the correct threshold is $(2N)^{1/2}$.
Let $a_1,\ldots,a_p$ be (not necessarily distinct) residues modulo $p$, such that there exists some $r$ so that if $S\subseteq [p]$ is non-empty and \[\sum_{i\in S}a_i\equiv 0\pmod{p}\] then $\lvert S\rvert=r$. Must there be at most two distinct residues amongst the $a_i$?
A question of Graham.
Let $A\subsetneq \mathbb{F}_p$. Is there some ordering $A=\{a_1,\ldots,a_k\}$ such that none of the consecutive sums $a_1+a_2+\cdots+a_i$ are $0$?
A question of Graham, who proved this is possible if $\lvert A\rvert=p-1$.
Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size $N$ and $A\subseteq G$ is a random set of size $k$ then, with probability $\geq 1/2$, all elements of $G$ can be written as $\sum_{x\in S}x$ for some $S\subseteq A$. Is \[f(N) \leq \log_2 N+o(\log\log N)?\]
Erdős and Rényi [ErRe65] proved that \[f(N) \leq \log_2N+O(\log\log N).\] Erdős believed improving this to $o(\log\log N)$ is impossible.
Show that \[R(3,k+1)-R(3,k)\to\infty\] as $k\to \infty$. Similarly, prove or disprove that \[R(3,k+1)-R(3,k)=o(k).\]
Show that if $G$ has $\binom{n}{2}$ edges then \[R(G) \leq R(n).\] More generally, if $G$ has $\binom{n}{2}+t$ edges with $t\leq n$ then \[R(G)\leq R(H)\] where $H$ is the graph formed by connected a new vertex to $t$ of the vertices of $K_n$.
In other words, are cliques extremal for Ramsey numbers. Asked by Erdős and Graham.

See also the entry in the graphs problem collection.

Is it true that if $G$ has $m$ edges then \[R(G) \leq 2^{O(m^{1/2})}?\]
If $T$ is a tree on $n$ vertices then \[R(T) \leq 2n-2.\]
Equality holds when $T$ is a star on $n$ vertices.

Implied by [548].

See also the entry in the graphs problem collection.

Let $n\geq k+1$. Every graph on $n$ vertices with at least $n(k-1)/2+1$ edges contains every tree on $k+1$ vertices.
A problem of Erdős and Sós. It can be easily proved by induction that every graph on $n$ vertices with at least $n(k-1)+1$ edges contains every tree on $k+1$ vertices.

Brandt and Dobson [BrDo96] have proved this for graphs of girth at least $5$. Wang, Li, and Liu [WLL00] have proved this for graphs whose complements have girth at least $5$. Saclé and Woznik [SaWo97] have proved this for graphs which contain no cycles of length $4$. Yi and Li [YiLi04] have proved this for graphs whose complements contain no cycles of length $4$.

Implies [547] and [557].

See also the entry in the graphs problem collection.

If $T$ is a tree which is a bipartite graph with $k$ vertices and $2k$ vertices in the other class then show that $R(T)=4k$.
It follows from results in [EFRS82] that $R(T)\geq 4k-1$.

See also the entry in the graphs problem collection.

Let $m_1\leq\cdots\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex class sizes $m_1,\ldots,m_k$ then prove that \[R(T,G)\leq (\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1.\]
Chvátal [Ch77] proved that $R(T,K_m)=(m-1)(n-1)+1$.

See also the entry in the graphs problem collection.

Prove that \[R(C_k,K_n)=(k-1)(n-1)+1\] for $k\geq n\geq 3$ (except when $n=k=3$).
Asked by Erdős, Faudree, Rousseau, and Schelp, who also ask for the smallest value of $k$ such that this identity holds (for fixed $n$). They also ask, for fixed $n$, what is the minimum value of $R(C_k,K_n)$?

This identity was proved for $k>n^2-2$ by Bondy and Erdős [BoEr73]. Nikiforov [Ni05] extended this to $k\geq 4n+2$.

Keevash, Long, and Skokan [KLS21] have proved this identity when $k\geq C\frac{\log n}{\log\log n}$ for some constant $C$, thus establishing the conjecture for sufficiently large $n$. They also prove that $R(C_k,K_n)\gg (k-1)(n-1)+1$ for all $3\leq k\leq (1-o(1))\frac{\log n}{\log\log n}$.

See also the entry in the graphs problem collection.

Determine \[R(C_4,S_n),\] where $S_n$ is the star on $n+1$ vertices.
It was shown in [BEFRS89] that \[n+\lceil\sqrt{n}\rceil+1\geq R(C_4,S_n)\geq n+\sqrt{n}-6n^{11/40}.\] Füredi (unpublished) has shown that $R(C_4,S_n)=n+\lceil\sqrt{n}\rceil$ for infinitely many $n$.

See also the entry in the graphs problem collection.

Let $R(3,3,n)$ denote the smallest integer $m$ such that if we $3$-colour the edges of $K_m$ then there is either a monochromatic triangle in one of the first two colours or a monochromatic $K_n$ in the third colour. Define $R(3,n)$ similarly but with two colours. Show that \[\frac{R(3,3,n)}{R(3,n)}\to \infty\] as $n\to \infty$.
A problem of Erdős and Sós. This was solved by Alon and Rödl [AlRo05], who in fact show that \[R(3,3,n)\asymp n^3(\log n)^{O(1)}\] (recalling that Shearer [Sh83] showed $R(3,n) \ll n^2/\log n$).

See also the entry in the graphs problem collection.

Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that \[\lim_{k\to \infty}\frac{R(C_{2n+1};k)}{R(K_3;k)}=0\] for any $n\geq 2$.
A problem of Erdős and Graham. The problem is open even for $n=2$.

See also the entry in the graphs problem collection.

Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine the value of \[R(C_{2n};k).\]
A problem of Erdős and Graham. Erdős [Er81c] gives the bounds \[k^{1+\frac{1}{2n}}\ll R(C_{2n};k)\ll k^{1+\frac{1}{n-1}}.\] Chung and Graham [ChGr75] showed that \[R(C_4;k)>k^2-k+1\] when $k-1$ is a prime power and \[R(C_4;k)\leq k^2+k+1\] for all $k$.

See also the entry in the graphs problem collection.

Let $R(G;3)$ denote the minimal $m$ such that if the edges of $K_m$ are $3$-coloured then there must be a monochromatic copy of $G$. Show that \[R(C_n;3) \leq 4n-3.\]
A problem of Bondy and Erdős. This inequality is best possible for odd $n$.

Luczak [Lu99] has shown that $R(C_n;3)\leq (4+o(1))n$ for all $n$, and in fact $R(C_n;3)\leq 3n+o(n)$ for even $n$.

Kohayakawa, Simonovits, and Skokan [KSS05] proved this conjecture when $n$ is sufficiently large and odd. Benevides and Skokan [BeSk09] proved that if $n$ is sufficiently large and even then $R(C_n;3)=2n$.

See also the entry in the graphs problem collection.

Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Is it true that \[R(T;k)=kn+O(1)\] for any tree $T$ on $n$ vertices?
A problem of Erdős and Graham. Implied by [548].

See also the entry in the graphs problem collection.

Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine \[R(K_{s,t};k)\] where $K_{s,t}$ is the complete bipartite graph with $s$ vertices in one component and $t$ in the other.
Chung and Graham [ChGr75] prove the bounds \[(2\pi\sqrt{st})^{\frac{1}{s+t}}\left(\frac{s+t}{e^2}\right)k^{\frac{st-1}{s+t}}\leq R(K_{s,t};k)\leq (t-1)(k+k^{1/s})^s.\] For example this implies that \[R(K_{3,3};k) \ll k^3.\] Using Turán numbers one can show that \[R(K_{3,3};k) \gg \frac{k^3}{(\log k)^3}.\]

See also the entry in the graphs problem collection.

Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges that is Ramsey for $G$.

If $G$ has $n$ vertices and maximum degree $d$ then prove that \[\hat{R}(G)\ll_d n.\]

A problem of Beck and Erdős. Beck [Be83b] proved this when $G$ is a path. Friedman and Pippenger [FrPi87] proved this when $G$ is a tree. Haxell, Kohayakawa, and Luczak [HKL95] proved this when $G$ is a cycle. An alternative proof when $G$ is a cycle (with better constants) was given by Javadi, Khoeini, Omidi, and Pokrovskiy [JKOP19].

This was disproved for $d=3$ by Rödl and Szemerédi [RoSz00], who constructed a graph on $n$ vertices with maximum degree $3$ such that \[\hat{r}(G)\gg n(\log n)^{c}\] for some absolute constant $c>0$. It is an interesting question how large $\hat{R}(G)$ can be if $G$ has maximum degree $3$ - the best known upper bound is $\hat{R}(G)\ll n^{8/5}$, proved by Conlon, Nenadov, and Trujić [CNT22].

See also the entry in the graphs problem collection.

Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.

Determine \[\hat{R}(K_{n,n}),\] where $K_{n,n}$ is the complete bipartite graph with $n$ vertices in each component.

We know that \[\frac{1}{60}n^22^n<\hat{R}(K_{n,n})< \frac{3}{2}n^32^n.\] The lower bound (which holds for $n\geq 6$) was proved by Erdős and Rousseau [ErRo93]. The upper bound was proved by Erdős, Faudree, Rousseau, and Schelp [EFRS78b] and Nešetřil and Rödl [NeRo78].

Conlon, Fox, and Wigderson [CFW23] have proved that, for any $s\leq t$, \[\hat{R}(K_{s,t})\gg s^{2-\frac{s}{t}}t2^s,\] and prove that when $t\gg s\log s$ we have $\hat{R}(K_{s,t})\asymp s^2t2^s$. They conjecture that this should hold for all $s\leq t$, and so in particular we should have $\hat{R}(K_{n,n})\asymp n^32^n$.

See also the entry in the graphs problem collection.

Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.

Let $F_1$ and $F_2$ be the union of stars. More precisely, let $F_1=\cup_{i\leq s} K_{1,n_i}$ and $F_2=\cup_{j\leq t} K_{1,m_j}$. Prove that \[\hat{R}(F_1,F_2) = \sum_{2\leq k\leq s+2}\max\{n_i+m_j-1 : i+j=k\}.\]

Burr, Erdős, Faudree, Rousseau, and Schelp [BEFRS78] proved this when all the $n_i$ are identical and all the $m_i$ are identical.

See also the entry in the graphs problem collection.

Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.

Prove that, for $r\geq 3$, \[\log_{r-1} R_r(n) \asymp_r n,\] where $\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm.

A problem of Erdős, Hajnal, and Rado [EHR65]. A generalisation of [564].

See also the entry in the graphs problem collection.

Let $F(n,\alpha)$ denote the largest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\subseteq [n]$ with $\lvert X\rvert\geq m$ contains more than $\alpha \binom{\lvert X\rvert}{2}$ many edges of each colour.

Prove that, for every $0\leq \alpha\leq 1/2$, \[F(n,\alpha)\sim c_\alpha\log n\] for some constant $c_\alpha$ depending only on $\alpha$.

It is easy to show that, for every $0\leq \alpha\leq 1/2$, \[F(n,\alpha)\asymp_\alpha \log n.\]

Note that when $\alpha=0$ this is just asking for a $2$-colouring of the edges of $K_n$ which contains no monochromatic clique of size $m$, and hence we recover the classical Ramsey numbers.

See also [161].

See also the entry in the graphs problem collection.

Let $R^*(G)$ be the induced Ramsey number: the minimal $m$ such that there is a graph $H$ on $m$ vertices such that any $2$-colouring of the edges of $H$ contains an induced monochromatic copy of $G$.

Is it true that \[R^*(G) \leq 2^{O(n)}\] for any graph $G$ on $n$ vertices?

A problem of Erdős and Rödl. Even the existence of $R^*(G)$ is not obvious, but was proved independently by Deuber [De75], Erdős, Hajnal, and Pósa [EHP75], and Rödl [Ro73].

Rödl [Ro73] proved this when $G$ is bipartite. Kohayakawa, Prömel, and Rödl [KPR98] have proved that \[R^*(G) < 2^{O(n(\log n)^2)}.\] An alternative (and more explicit) proof was given by Fox and Sudakov [FoSu08].

See also the entry in the graphs problem collection.

Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then \[R(G,H)\ll m?\]
In other words, is $G$ Ramsey size linear? This fails for a graph $G$ with $n$ vertices and $2n-2$ edges (for example with $H=K_n$). Erdős, Faudree, Rousseau, and Schelp [EFRS93] have shown that any graph $G$ with $n$ vertices and at most $n+1$ edges is Ramsey size linear.

Implies [567].

See also the entry in the graphs problem collection.

Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ has $m$ edges and no isolated vertices, then \[R(G,H)\ll m?\]
In other words, is $G$ Ramsey size linear? A special case of [566]. In [Er95] Erdős specifically asks about the case $G=K_{3,3}$.

The graph $H_5$ can also be described as $K_4^*$, obtained from $K_4$ by subdividing one edge. ($K_4$ itself is not Ramsey size linear, since $R(4,n)\gg n^{3-o(1)}$, see [166].) Bradać, Gishboliner, and Sudakov [BGS23] have shown that every subdivision of $K_4$ on at least $6$ vertices is Ramsey size linear, and also that $R(H_5,H) \ll m$ whenever $H$ is a bipartite graph with $m$ edges and no isolated vertices.

See also the entry in the graphs problem collection.

Let $G$ be a graph such that $R(G,T_n)\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices, \[R(G,H)\ll m?\]
In other words, is $G$ Ramsey size linear?

See also the entry in the graphs problem collection.

Let $k\geq 1$. What is the best possible $c_k$ such that \[R(C_{2k+1},H)\leq c_k m\] for any graph $H$ on $m$ edges without isolated vertices?
Let $k\geq 3$. Is it true that, for any graph $H$ on $m$ edges without isolated vertices, \[R(C_k,H) \leq 2m+\left\lceil\frac{k-1}{2}\right\rceil?\]
This was proved for even $k$ by Erdős, Faudree, Rousseau, and Schelp [EFRS93]. It was proved for $k=3$ by Sidorenko [Si93].

See also the entry in the graphs problem collection.

Show that for any rational $\alpha \in (1,2)$ there exists a bipartite graph $G$ such that \[\mathrm{ex}(n;G)\asymp n^{\alpha}.\] Conversely, if $G$ is bipartite then must there exist some rational $\alpha$ such that\[\mathrm{ex}(n;G)\asymp n^{\alpha}?\]
A problem of Erdős and Simonovits.

See also the entry in the graphs problem collection.

Show that for $k\geq 3$ \[\mathrm{ex}(n;C_{2k})\gg n^{1+\frac{1}{k}}.\]
It is easy to see that $\mathrm{ex}(n;C_{2k+1})=\lfloor n^2/4\rfloor$ for any $k\geq 1$ (and $n>2k+1$) (since no bipartite graph contains an odd cycle). It is also known that $\mathrm{ex}(n;C_4)\asymp n^{3/2}$.

Erdős [Er65] and Bondy and Simonovits [BoSi74] showed that \[\mathrm{ex}(n;C_{2k})\ll kn^{1+\frac{1}{k}}.\]

Benson [Be66] has proved this conjecture for $k=3$ and $k=5$. Lazebnik, Ustimenko, and Woldar [LUW95] have shown that, for arbitrary $k\geq 3$, \[\mathrm{ex}(n;C_{2k})\gg n^{1+\frac{2}{3k-3+\nu}},\] where $\nu=0$ if $k$ is odd and $\nu=1$ if $k$ is even. See [LUW99] for further history and references.

See also the entry in the graphs problem collection.

Is it true that \[\mathrm{ex}(n;\{C_3,C_4\})=(n/2)^{3/2}+O(n)?\]
A problem of Erdős and Simonovits, who proved that \[\mathrm{ex}(n;\{C_4,C_5\})=(n/2)^{3/2}+O(n).\]

See also [574] and the entry in the graphs problem collection.

Is it true that, for $k\geq 2$, \[\mathrm{ex}(n;\{C_{2k-1},C_{2k}\})=(1+o(1))(n/2)^{1+\frac{1}{k}}.\]
A problem of Erdős and Simonovits.

See also [573] and the entry in the graphs problem collection.

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 $\mathcal{F}$, if there is a bipartite graph in $\mathcal{F}$ then there exists some bipartite $G\in\mathcal{F}$ such that \[\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?\]

A problem of Erdős and Simonovits.

See also [180] and the entry in the graphs problem collection.

Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of \[\mathrm{ex}(n;Q_k).\]
Erdős and Simonovits [ErSi70] proved that \[(\tfrac{1}{2}+o(1))n^{3/2}\leq \mathrm{ex}(n;Q_3) \ll n^{8/5}.\] A theorem of Sudakov and Tomon [SuTo22] implies \[\mathrm{ex}(n;Q_k)=o(n^{2-\frac{1}{k}}).\] Janzer and Sudakov [JaSu24] have improved this to \[\mathrm{ex}(n;Q_k)\ll_k n^{2-\frac{1}{k-1}+\frac{1}{(k-1)2^{k-1}}}.\] See also the entry in the graphs problem collection.
If $G$ is a graph with $4k$ vertices and minimum degree at least $2k$ then $G$ contains $k$ vertex-disjoint $4$-cycles.
A conjecture of Erdős and Faudree. Proved by Wang [Wa10].
If $G$ is a random graph on $2^d$ vertices, including each edge with probability $1/2$, then $G$ almost surely contains a copy of $Q_d$ (the $d$-dimensional hypercube with $2^d$ vertices and $d2^{d-1}$ many edges).
A conjecture of Erdős and Bollobás. Solved by Riordan [Ri00], who in fact proved this with any edge-probability $>1/4$, and proves that the number of copies of $Q_d$ is normally distributed.

See also the entry in the graphs problem collection.

Let $\epsilon>0$ and $n$ be sufficiently large. Show that, if $G$ is a graph on $n$ vertices which does not contain $K_{2,2,2}$ and $G$ has at least $\epsilon n^2$ many edges, then $G$ contains an independent set on $\gg_\epsilon n$ many vertices.
A problem of Erdős, Hajnal, Sós, and Szemerédi.

See also the entry in the graphs problem collection.

Let $G$ be a graph on $n$ vertices such that at least $n/2$ vertices have degree at least $n/2$. Must $G$ contain every tree on at most $n/2$ vertices?
A conjecture of Erdős, Füredi, Loebl, and Sós. Ajtai, Komlós, and Szemerédi [AKS95] proved an asymptotic version, where at least $(1+\epsilon)n/2$ vertices have degree at least $(1+\epsilon)n/2$ (and $n$ is sufficiently large depending on $\epsilon$).

Komlós and Sós conjectured the generalisation that if at least $n/2$ vertices have degree at least $k$ then $G$ contains any tree with $k$ vertices.

See also the entry in the graphs problem collection.

Let $f(m)$ be the maximal $k$ such that a triangle-free graph on $m$ edges must contain a bipartite graph with $k$ edges. Determine $f(m)$.
Resolved by Alon [Al96], who showed that there exist constants $c_1,c_2>0$ such that \[\frac{m}{2}+c_1m^{4/5}\leq f(m)\leq \frac{m}{2}+c_2m^{4/5}.\]

See also the entry in the graphs problem collection.

Every connected graph on $n$ vertices can be partitioned into at most $\lceil n/2\rceil$ edge-disjoint paths.
A problem of Erdős and Gallai. Lovász [Lo68] proved that every graph on $n$ vertices can be partitioned into at most $\lfloor n/2\rfloor$ edge-disjoint paths and cycles. Chung [Ch78] proved that every connected graph on $n$ vertices can be partitioned into at most $\lceil n/2\rceil$ edge-disjoint trees. Pyber [Py96] has shown that every connected graph on $n$ vertices can be covered by at mst $n/2+O(n^{3/4})$ paths.

Hajos [Lo68] has conjectured that if $G$ has all degrees even then $G$ can be partitioned into at most $\lfloor n/2\rfloor$ edge-disjoint cycles.

See also [184] and the entry in the graphs problem collection.

Let $G$ be a graph with $n$ vertices and $\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\subseteq G$ such that
  • $H_1$ has $\gg \delta^3n^2$ edges and every two edges in $H_1$ are contained in a cycle of length at most $6$, and furthermore if two edges share a vertex they are on a cycle of length $4$, and
  • $H_2$ has $\gg \delta^2n^2$ edges and every two edges in $H_2$ are contained in a cycle of length at most $8$.
A problem of Erdős, Duke, and Rödl. Duke and Erdős [DuEr83], who proved the first if $n$ is sufficiently large depending on $\delta$. The real challenge is to prove this when $\delta=n^{-c}$ for some $c>0$. Duke, Erdős, and Rödl [DER84] proved the first statement with a $\delta^5$ in place of a $\delta^3$.

Fox and Sudakov [FoSu08b] have proved the second statement when $\delta >n^{-1/5}$.

See also the entry in the graphs problem collection.

What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same vertex set?
Pyber, Rödl, and Szemerédi [PRS95] constructed such a graph with $\gg n\log\log n$ edges.

Chakraborti, Janzer, Methuku, and Montgomery [CJMM24] have shown that such a graph can have at most $n(\log n)^{O(1)}$ many edges. Indeed, they prove that there exists a constant $C>0$ such that for any $k\geq 2$ there is a $c_k$ such that if a graph has $n$ vertices and at least $c_kn(\log n)^{C}$ many edges then it contains $k$ pairwise edge-disjoint cycles with the same vertex set.