Problem 142

Erdős Problem #142: declared status 'open'. Formalized: yes. Let $r_k(N)$ be the largest possible size of a subset of $\{1,\ldots,N\}$ that does not contain any non-trivial $k$-term arithmetic progression. Prove an asymptotic formula for $r_k(N)$. Current best: The best known upper bounds for $r_k(N)$ are due to Kelley and Meka [KeMe23] for $k=3$, Green and Tao [GrTa17] for $k=4$, and Leng, Sah, and Sawhney [LSS24] for $k\geq 5$. An asymptotic formula is still far out of reach, even for $k=3$. Prize: $10000. OEIS: A003002, A003003, A003004, A003005. Tags: additive combinatorics, arithmetic progressions.