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Claim. Nat Sothanaphan, A sieve refinement for van der Waerden differences, a five-page note dated 11 April 2026 in print and linked from the site's thread for Problem 138 on 10 April 2026. Its AI disclosure says it was produced with GPT-5.4 Thinking. For $r$ colors let $W_r(k)$ be the $r$-color van der Waerden number, so that $W(k)=W_2(k)$. Theorem 4: for all $k,r\ge2$,

$$W_r(k+1)-W_r(k)\ge k+\min\{k,F(r)\}+1$$

for an explicit computable function $F$ with $F(r)\ge r-2$ (Corollary 5) and $F(r)=(e^{\gamma}+o(1))\,r\log\log r$ (Proposition 6). At $r=2$ the bound is $W(k+1)-W(k)\ge k+1$, one more than the bound $W(k+1)\ge W(k)+k$ on the DeepMind claim page. The note proves its bound in full by the greedy extension of a progression-free coloring that it credits, for the method, to the DeepMind argument and to the curator's comment on the thread. The argument was not reconstructed in this corpus.

Submission note. Posted to the site's forum by Nat Sothanaphan on 10 April 2026:

GPT-5.4 Thinking and I in these notes have refined the difference bound to:

$$> W_r(k+1) - W_r(k) \ge k + \min(k, F(r)) + 1, >$$

where $F(r) = \Theta(r \log \log r)$ is an explicit function. So for large $r$ and large $k$ depending on $r$, the difference is at least $k + \Theta(r \log \log r)$.

Covers. The question $W(k+1)-W(k)\to\infty$ of [Er81], through $W(k+1)-W(k)\ge k+1$. Not covered: the problem's request and its example question $W(k)^{1/k}\to\infty$, the quotient question $W(k+1)/W(k)\to\infty$, and the bounds for $r\ge3$ colors, which concern $W_r$ and not the problem's two-color number.

Depends on. No page of this wiki.

Standing. Claimed: the note has no journal record and no outside review, the site's commentary does not mention it, and the site labels the problem OPEN. No Lean formalization of the note's theorem is known, and this corpus has built nothing, so the page lists no evidence.