A published paper or arXiv preprint answers the main question of Erdős Problem #3 (theorem erdos_3 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):

If A \subset \mathbb{N} has \sum_{n \in A}\frac 1 n = \infty, then must A contain arbitrarily long arithmetic progressions? Yes. This was proved by an internal OpenAI model in September 2026 [OAI26]; see the linked Lean proof.

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

• [BlSi20] Bloom, Thomas F. and Sisask, Olof, Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. arXiv:2007.03528 (2020).
• [Go01] Gowers, W. T., A new proof of Szemerédi's theorem. Geom. Funct. Anal. (2001), 465-588.
• [GrTa17] Green, Ben and Tao, Terence, *New bounds for Szemerédi's theorem, III: a polylogarithmic bound for r_4(N)*. Mathematika (2017).
• [KeMe23] Kelley, Zander and Meka, Raghu, Strong bounds for 3-progressions. 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) (2023).
• [LSS24] Leng, James, Sah, Ashwin and Sawhney, Mehtaab, Improved bounds for Szemerédi's theorem. arXiv:2402.17995 (2024).
• [OAI26] OpenAI, Quasipolynomial Bounds for Arithmetic Progressions. OpenAI Math Release preprint (2026).