SOLVED - $500
Let $r_3(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain a non-trivial $3$-term arithmetic progression. Prove that $r_3(N)\ll N/(\log N)^C$ for every $C>0$.
Proved by Kelley and Meka
[KeMe23]. In
[ErGr80] and
[Er81] it is conjectured that this holds for every $k$-term arithmetic progression.
See also [3].
