Open-problem statements, with their sources attached.

$W(3, 31) > 930$ from [AKS14, Table 3].

$W(3, 32) > 1006$ from [AKS14, Table 3].

$W(3, 33) > 1063$ from [AKS14, Table 3].

$W(3, 34) > 1143$ from [AKS14, Table 3].

$W(3, 35) > 1204$ from [AKS14, Table 3].

$W(3, 36) > 1257$ from [AKS14, Table 3].

$W(3, 37) > 1338$ from [AKS14, Table 3].

$W(3, 38) > 1378$ from [AKS14, Table 3].

$W(3, 39) > 1418$ from [AKS14, Table 3].

Does there exist a Lipschitz function $f : \mathbb{N} \to \mathbb{Z}$ whose graph $\Gamma = \{(n, f(n)) : n \in \mathbb{N}\} \subseteq \mathbb{Z}^2$ is free of 3-term progressions?

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

From [Ruzsa] $f(N) \gg N^{1/2}$.

From [Schoen and Sisask] $f(N) \ll N \cdot e^{-c(\log N)^{1/7}}$.

$f(N) \gg N \cdot e^{-c(\log N)^{1/7}}$.

From [Yufei Zhao]: Is there a subset of $\{1, \ldots, N\}$ of size $N^{1/3 - o(1)}$ with no nontrivial solutions to $x + 2y + 3z = x' + 2y' + 3z'$?

Suppose that $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$. Is it true that there are $\gg_\alpha |G|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$ all lie in $A$?

Note: A is taken as $\alpha$-dense, i.e. $|A| \ge \alpha |G|^2$ [Au16, Question 2]

[Ma21] showed that $3.13 \leq C$.

[Ma21] showed that $C \leq 4$.

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: $\left({1}{3} + o(1)\right) n^2$.