Problem Statement
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 $C>0$.
Categories:
Additive Combinatorics Arithmetic Progressions
Progress
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].
Source: erdosproblems.com/140 | Last verified: January 13, 2026