A published paper or arXiv preprint answers the main question of Erdős Problem #817 (theorem erdos_817 in formal-conjectures) in full (proof, disproof, or the exact value asked for) with no hypotheses beyond the problem's own.

The problem, as stated in formal-conjectures (Apache 2.0):

Let k \geq 3. Define g_k(n) to be the minimal N such that \{1, ..., N\} contains some A of size |A| = n such that \langle A\rangle = \left\{\sum_{a \in A} \epsilon_a a : \epsilon_a \in\{0, 1\}\right\} contains no non-trivial k-term arithmetic progression. Estimate g_k(n). In particular, is it true that g_3(n) \gg 3^n The answer is no: Costa [Co26, Corollary 1.2] proved that \liminf_{n \to \infty} g_3(n)/3^n = 0.

References formal-conjectures lists for this problem (a starting point, not evidence):

• [Co26] Costa, S., A negative answer to the Erdős–Sárközy question. arXiv:2609.06303 (2026).