Claims

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2026_06_21_kitamura: Kenta Kitamura's Lean 4 development proves N(k,2) <= floor(2(k^2-1)(k-1) (2k+1)/(k-3)) + 1 for every k >= 5, so N(k,2) = O(k^3) and the displayed question N(k,2) <= C^k has the answer yes; prepared with Codex and ChatGPT.

2026_06_23_kitamura: Kenta Kitamura's Lean 4 development proves N(k, sqrt k) <= floor((4/3) (k^2-k+1)(k-1)(k(k-1)+1)) + 1 for every k >= 2, so N(k, sqrt k) = O(k^5) and the displayed question has the answer yes; prepared with Codex and ChatGPT.

2026_09_23_openai: The OpenAI release's superexponential lower bound for van der Waerden numbers, read through the site's identity N(k,k) = W(k), makes the first displayed question false at c = 1; a release preprint, claimed, not accepted.

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