Unsolved Problems

Fix an integer $d$. What is the largest sum-free subset of $[N]^d$?...

Let $S \subset \mathbb{N}$ be random. Under what conditions is Roth's theorem for progressions of length 3 true with common differences in $S$?...

Find reasonable bounds for the maximal density of a set $A \subset \{1, \ldots, N\}$ not containing a 3-term progression with common difference a squa...

Does there exist a Lipschitz function $f : \mathbb{N} \to \mathbb{Z}$ whose graph $\Gamma = \{(n, f(n)) : n \in \mathbb{Z}\} \subset \mathbb{Z}^2$ is ...

What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?...

Suppose $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$. Are there $\gg_\alpha |G|^3$ triples $x, y, g$ such tha...

Suppose that $A \subset \mathbb{F}_2^n$ has density $\alpha$. What is the largest size of coset guaranteed to be contained in $2A$?...

Consider a set $S \subset [N]^3$ with the property that any two distinct elements $s, s'$ of $S$ are comparable (in the coordinatewise partial order)....

In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \times n$ chessboard?...

If $\{1, \dots, N\}$ is $r$-coloured, then for $N \geq N_0(r)$ there exist integers $x, y \geq 3$ such that $x+y$ and $xy$ have the same colour. Find ...

If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set $\{0, 1, 3\}$ that $A$ can contain?...

Suppose $X, Y$ are finitely-supported independent random variables taking integer values such that $X + Y$ is uniformly distributed on its range. Are ...

Let $p$ be a prime and let $A \subset \mathbb{Z}/p\mathbb{Z}$ be a set of size $\sqrt{p}$. Is there a dilate of $A$ with a gap of length $100\sqrt{p}$...

Suppose $A \subset [N]$ has size $\geq c\sqrt{N}$ and representation function $r_A(n) \leq r$ for all $n$. What can be said about the structure of $A$...

If $A \subset \mathbb{Z}/p\mathbb{Z}$ is random with $|A| = \sqrt{p}$, can we almost surely cover $\mathbb{Z}/p\mathbb{Z}$ with $100\sqrt{p}$ translat...

In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \times n$ chessboard?...