Open-problem statements, with their sources attached.

If $\{1, \ldots, N\}$ is $r$-coloured then, for $N \geqslant N_0(r)$, there are integers $x, y \geqslant 3$ such that $x + y, xy$ have the same colour.

Find reasonable bounds for $N_0(r)$. The goal is to improve upon the Green-Sawhney bound.

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

Conjectured in [Aa19] p.579: $\left({1}{3} + o(1)\right) n^2$.

Conjecture p.579 in [Aa19]: $\left({1}{3} + o(1)\right) n^2$.

For which values of $k$ is the following true: whenever we partition $[N] = A_1 \cup \dots \cup A_k$, $\left|\bigcup^k_{i=1} (A_i \hat{+} A_i)\right| \geq \frac{1}{10} N$?

We conjecture that the best-known upper bound can be lowered.

We conjecture that the best-known lower bound can be raised.

The analogous problem in $\mathbb{F}^n_p$ remains open. [Gr24]

What is the size of the smallest set $A \subset \mathbb{Z} / p\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation?

Propose a better lower bound along primes.

Propose a better upper bound along primes.

Can we improve the lower bound $N^{1/2} + O(1)$, at least for infinitely many $N$?

Can we improve the lower bound $N^{1/2} + O(1)$, for all sufficiently large $N$?

Can we improve the upper bound $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25], at least for infinitely many $N$?

Can we improve the upper bound $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25], for all sufficiently large $N$?

It is not known whether or not there exists a Sidon subset of $\mathbb{Z}/p\mathbb{Z}$ of size $(1 + o(1))\sqrt{p}$, for all $p$ [Gr24].

It is not known whether, if $G$ is an abelian group of size $n$, there always exists a Sidon subset of $G$ of size $0.01\sqrt{n}$ [Gr24].

Another very nice old problem is whether there is a Sidon subset of $\{0, 1\}^n$ of size $N^{0.51}$, where $N = 2^n$ [Gr24].

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

Even what happens in the regime $\omega(p) \sim 10 \log p$ is unclear [Gr24].

Are there infinitely many $q$ for which there is a set $A \subset \mathbb{Z}/q\mathbb{Z}$, $|A| = (\sqrt{2} + o(1))q^{1/2}$, with $A + A = \mathbb{Z}/q\mathbb{Z}$? [Gr24]