SOLVED - $1000
Let $r_k(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain a non-trivial $k$-term arithmetic progression. Prove that $r_k(N)=o(N)$.
Proved by Szemerédi
[Sz74]. The best known bounds are due to Kelley and Meka
[KeMe23] for $k=3$ (with further slight improvements in
[BlSi23]), Green and Tao
[GrTa17] for $k=4$, and Leng, Sah, and Sawhney
[LSS24] for $k\geq 5$.
See also [3].
