OPEN - $250

Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$
\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?\]

The sum-product problem. Erdős and Szemerédi [ErSz83] proved a lower bound of $\lvert A\rvert^{1+c}$ for some constant $c>0$, and an upper bound of
\[\lvert A\rvert^2 \exp\left(-c\frac{\log\lvert A\rvert}{\log\log \lvert A\rvert}\right)\]
for some constant $c>0$. The lower bound has been improved a number of times. The current record is
\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{1558}{1167}-o(1)}\]
due to Rudnev and Stevens [RuSt22] (note $1558/1167=1.33504\cdots$).

There is likely nothing special about the integers in this question, and indeed Erdős and Szemerédi also ask a similar question about finite sets of real or complex numbers. The current best bound for sets of reals is the same bound of Rudnev and Stevens above. The best bound for complex numbers is \[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{5}{4}},\] due to Solymosi [So05].

One can in general ask this question in any setting where addition and multiplication are defined (once one avoids any trivial obstructions such as zero divisors or finite subfields). For example, it makes sense for subsets of finite fields. The current record is that if $A\subseteq \mathbb{F}_p$ with $\lvert A\rvert <p^{5/8}$ then \[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{11}{9}+o(1)},\] due to Rudnev, Shakan, and Shkredov [RSS20].

There is also a natural generalisation to higher-fold sum and product sets. For example, in [ErSz83] (and in [Er91]) Erdős and Szemerédi also conjecture that for any $m\geq 2$ and finite set of integers $A$ \[\max( \lvert mA\rvert,\lvert A^m\rvert)\gg \lvert A\rvert^{m-o(1)}.\] See [53] for more on this generalisation and [808] for a stronger form of the original conjecture. See also [818] for a special case.

SOLVED - $250

A set of integers $A$ is Ramsey $r$-complete if, whenever $A$ is $r$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of $A$. Prove any non-trivial bounds about the growth rate of such an $A$ for $r>2$.

A paper of Burr and Erdős [BuEr85] proves both upper and lower bounds for $r=2$, showing that there exists some $c>0$ such that it cannot be true that
\[\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2\]
for all large $N$, and also constructing a Ramsey $2$-complete $A$ such that for all large $N$
\[\lvert A\cap \{1,\ldots,N\}\rvert \ll (\log N)^3.\]
Burr has shown that the sequence of $k$th powers is Ramsey $r$-complete for every $r,k\geq 1$.

Solved by Conlon, Fox, and Pham [CFP21], who constructed for every $r\geq 2$ an $r$-Ramsey complete $A$ such that for all large $N$ \[\lvert A\cap \{1,\ldots,N\}\rvert \ll r(\log N)^2,\] and showed that this is best possible, in that there exists some constant $c>0$ such that if $A\subset \mathbb{N}$ satisfies \[\lvert A\cap \{1,\ldots,N\}\rvert \leq cr(\log N)^2\] for all large $N$ then $A$ cannot be $r$-Ramsey complete.

OPEN - $250

Find the value of $\lim_{k\to \infty}R(k)^{1/k}$.

Erdős offered \$100 for just a proof of the existence of this constant, without determining its value. He also offered \$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. Erdős proved
\[\sqrt{2}\leq \liminf_{k\to \infty}R(k)^{1/k}\leq \limsup_{k\to \infty}R(k)^{1/k}\leq 4.\]
The upper bound has been improved to $4-\tfrac{1}{128}$ by Campos, Griffiths, Morris, and Sahasrabudhe [CGMS23].

This problem is #3 in Ramsey Theory in the graphs problem collection.

SOLVED - $250

If $G$ is bipartite then $\mathrm{ex}(n;G)\ll n^{3/2}$ if and only $G$ is $2$-degenerate, that is, $G$ contains no induced subgraph with minimal degree at least 3.

Conjectured by Erdős and Simonovits. Erdős offered \$250 for a proof and \$100 for a counterexample. Disproved by Janzer [Ja21], who constructed, for any $\epsilon>0$, a $3$-regular bipartite graph $H$ such that
\[\mathrm{ex}(n;H)\ll n^{\frac{4}{3}+\epsilon}.\]

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

OPEN - $250

Let $a,b,c$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\geq 0$), none of which divide any other?

Conjectured by Erdős and Lewin [ErLe96], who (among other related results) prove this when $a=3$, $b=5$, and $c=7$.

In [Er92b] Erdős wrote 'last year I made the following silly conjecture': every integer $n$ can be written as the sum of distinct integers of the form $2^k3^l$, none of which divide any other. 'I mistakenly thought that this was a nice and difficult conjecture but Jansen and several others found a simple proof by induction.'

In [Er92b] Erdős makes the stronger conjecture (for $a=2$, $b=3$, and $c=5$) that, for any $\epsilon>0$, all large integers $n$ can be written as the sum of distinct integers $b_1<\cdots <b_t$ of the form $2^k3^l5^m$ where $b_t<(1+\epsilon)b_1$.

See also [845].

OPEN - $250

Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a\neq b\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\log n\to\infty$?

Investigated by Erdős and Turán [ErTu34] (prompted by a question of Lázár and Grünwald) in their first joint paper, where they proved that
\[\log n \ll f(n) \ll n/\log n\]
(the upper bound is trivial, taking $A=\{1,\ldots,n\}$). Erdős says that $f(n)=o(n/\log n)$ has never been proved, but perhaps never seriously attacked.

OPEN - $250

Let $G$ be a graph with $n$ vertices such that every subgraph on $\geq n/2$ vertices has more than $n^2/50$ edges. Must $G$ contain a triangle?

A problem of Erdős and Rousseau. The constant $50$ would be best possible as witnessed by a blow-up of $C_5$ or the Petersen graph. Krivelevich [Kr95] has proved this with $n/2$ replaced by $3n/5$ (and $50$ replaced by $25$).

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

SOLVED - $250

Let $A\subset \mathbb{R}^2$ be a set of $n$ points such that any subset of size $4$ determines at least $5$ distinct distances. Must $A$ determine $\gg n^2$ many distances?

Erdős also makes the even stronger conjecture that $A$ must contain $\gg n$ many points such that all pairwise distances are distinct.

Answered in the negative by Tao [Ta24c], who proved that for any large $n$ there exists a set of $n$ points in $\mathbb{R}^2$ such that any four points determine at least give distinct distances, yet there are $\ll n^2/\sqrt{\log n}$ distinct distances in total. Tao discusses his solution in a blog post.

SOLVED - $250

The density of integers which have two divisors $d_1,d_2$ such that $d_1<d_2<2d_1$ exists and is equal to $1$.

It is known that there exists some constant $c>0$ such that for large $k$
\[c\frac{k^2}{\log k}\leq R(3,k) \leq (1+o(1))\frac{k^2}{\log k}.\]
The lower bound is due to Kim [Ki95], the upper bound is due to Shearer [Sh83], improving an earlier bound of Ajtai, Komlós, and Szemerédi [AjKoSz80]. The lower bound has been improved to
\[\left(\frac{1}{4}-o(1)\right)\frac{k^2}{\log k}\]
independently by Bohman and Keevash [BoKe21] and Pontiveros, Griffiths and Morris [PGM20]. The latter collection of authors conjecture that this lower bound is the true order of magnitude.

See also [544].

Spencer [Sp77] proved
\[R(4,k) \gg (k\log k)^{5/2}.\]
Ajtai, Komlós, and Szemerédi [AKS80] proved
\[R(4,k) \ll \frac{k^3}{(\log k)^2}.\]
This is true, and was proved by Mattheus and Verstraete [MaVe23], who showed that
\[R(4,k) \gg \frac{k^3}{(\log k)^4}.\]

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

OPEN - $250

Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic triangle. Determine
\[\lim_{k\to \infty}R(3;k)^{1/k}.\]

Erdős offers \$100 for showing that this limit is finite. An easy pigeonhole argument shows that
\[R(3;k)\leq 2+k(R(3;k-1)-1),\]
from which $R(3;k)\leq \lceil e k!\rceil$ immediately follows. The best-known upper bounds are all of the form $ck!+O(1)$, and arise from this type of inductive relationship and computational bounds for $R(3;k)$ for small $k$. The best-known lower bound (coming from lower bounds for Schur numbers) is due to Exoo [Ex94],
\[R(3;k) \gg (321)^{k/5}.\]

See also [483].

SOLVED - $250

Let $\alpha$ be the infinite ordinal $\omega^\omega$. Is it true that in any red/blue colouring of the edges of $K_\alpha$ there is either a red $K_\alpha$ or a blue $K_3$?

A problem of Erdős and Rado. For comparison, Specker [Sp57] proved this property holds when $\alpha=\omega^2$ and false when $\alpha=\omega^n$ for $3\leq n<\omega$.

This is true, and was proved by Chang [Ch72]. Milner modified his proof to prove that this remains true if we replace $K_3$ by $K_m$ for all finite $m<\omega$ (a shorter proof was found by Larson [La73]).

OPEN - $250

Let $\alpha$ be the infinite ordinal $\omega^{\omega^2}$. Is it true that in any red/blue colouring of the edges of $K_\alpha$ there is either a red $K_\alpha$ or a blue $K_3$?

OPEN - $250

Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?

A problem of Erdős and Hajnal. Folkman [Fo70] and Nešetřil and Rödl [NeRo75] have proved that for every $n\geq 1$ there is a graph $G$ which contains no $K_4$ and is not the union of $n$ triangle-free graphs.

SOLVED - $250

For a set of $n$ points $P\subset \mathbb{R}^2$ let $\ell_1,\ldots,\ell_m$ be the lines determined by $P$, and let $A=\{\lvert \ell_1\cap P\rvert,\ldots,\lvert \ell_m\cap P\rvert\}$.

Let $F(n)$ count the number of possible sets $A$ that can be constructed this way. Is it true that \[F(n) \leq \exp(O(\sqrt{n}))?\]

Erdős writes it is 'easy to see' that this bound would be best possible. This was proved by Szemerédi and Trotter [SzTr83].

SOLVED - $250

Let $r\geq 1$ and define $T(n,r)$ to be maximal such that there exists a family $\mathcal{F}$ of subsets of $\{1,\ldots,n\}$ of size $T(n,r)$ such that $\lvert A\cap B\rvert\neq r$ for all $A,B\in \mathcal{F}$.

Estimate $T(n,r)$ for $r\geq 2$. In particular, is it true that for every $\epsilon>0$ there exists $\delta>0$ such that for all $\epsilon n<r<(1/2-\epsilon) n$ we have \[T(n,r)<(2-\delta)^n?\]

It is trivial that $T(n,0)=2^{n-1}$. Frankl and Füredi [FrFu84] proved that, for fixed $r$ and $n$ sufficiently large in terms of $r$, the maximal $T(n,r)$ is achieved by taking
\[\mathcal{F} = \left\{ A\subseteq \{1,\ldots,n\} : \lvert A\rvert> \frac{n+r}{2}\textrm{ or }\lvert A\rvert < r\right\}\]
when $n+r$ is odd, and
\[\mathcal{F} = \left\{ A\subseteq \{1,\ldots,n\} : \lvert A\backslash \{1\}\rvert\geq \frac{n+r}{2}\textrm{ or }\lvert A\rvert < r\right\}\]
when $n+r$ is even. (Frankl [Fr77b] had earlier proved this for $r=1$ and all $n$.)

An affirmative answer to the second question implies that the chromatic number of the unit distance graph in $\mathbb{R}^n$ (with two points joined by an edge if the distance between them is $1$) grows exponentially in $n$, which was proved by alternative methods by Frankl and Wilson [FrWi81] - see [704].

The answer to the second question is yes, proved by Frankl and Rödl [FrRo87].

See also [702].