A published paper or arXiv preprint answers the main question of Erdős Problem #138 (theorem erdos_138 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):

In [Er80] Erdős asks whether \lim_{k \to \infty} (W(k))^{1/k} = \infty Yes. This was proved by an internal OpenAI model in September 2026 [OAI26], which shows W(k) > k^{k/100000} for all large k; see the linked Lean proof.

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

• [Be68] Berlekamp, E. R., A construction for partitions which avoid long arithmetic progressions. Canad. Math. Bull. (1968), 409-414.
• [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
• [Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
• [Go01] Gowers, W. T., A new proof of Szemerédi's theorem. Geom. Funct. Anal. (2001), 465-588.
• arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
• [CFS26] Campos, M., Fox, J. and Schildkraut, C., A new lower bound for two-color van der Waerden numbers. arXiv:2608.20824 (2026).
• [OAI26] OpenAI, Quantitative Superexponential Bounds for van der Waerden Numbers. OpenAI Math Release preprint (2026).