3 solved out of 12 shown

$250

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

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

$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.

Erdős offers \$100 for just a proof of the existence of this constant, without determining its value. He also offers \$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. Erdős proved
\[\sqrt{2}\leq \liminf_{k\to \infty}R(k)^{1/k}\leq \limsup_{k\to \infty}R(k)^{1/k}\leq 4.\]
The upper bound has been improved to $4-\tfrac{1}{128}$ by Campos, Griffiths, Morris, and Sahasrabudhe [CaGrMoSa23].

$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 $c=7$.

$250

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

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

$250

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

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

$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.

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

Spencer [Sp77] proved
\[R(4,k) \gg (k\log k)^{5/2}.\]
Ajtai, Komlós, and Szemerédi [AjKoSz80] proved
\[R(4,k) \ll \frac{k^3}{(\log k)^2}.\]
This is true, and was proved by Mattheus and Verstraete [MaVe23], who showed that
\[R(4,k) \gg \frac{k^3}{(\log k)^4}.\]

$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].