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Claim. For the two-color van der Waerden number $W(k)$ of Problem 138,

$$W(k)\ge(1-o(1))\,k\,2^{k-1}$$

for all positive integers $k$: Theorem 1 of M. Campos, J. Fox and C. Schildkraut, A new lower bound for two-color van der Waerden numbers, arXiv:2608.20824 (v1 of 21 August 2026, 5 pages). Berlekamp's bound $W(p+1)\ge p2^p$ [Be68] is the same statement when $k-1$ is prime; the paper's coloring for general $k$ is a product of Berlekamp's colorings for several primes, passed from cyclic groups to intervals through the cyclic van der Waerden number. The abstract says the result verifies a conjecture of Erdős. The paper's statement on AI use says that the coloring and an initial proof were generated by a query to ChatGPT 5.6 Sol Pro, and that the proof presented, while drawing on that output, was written by the authors as a modification of their exposition of Berlekamp's proof. The paper was posted to the site's thread on 2026-08-24.

Covers. The general lower bound: $W(k)\ge(1-o(1))k2^{k-1}$ improves Kozik and Shabanov's $W(k)\gg2^k$ [KoSh16], the record the problem page cites; it gives $W(k)/2^k\to\infty$, the question Erdős asked in [Er80] and the commentary records. Not covered: $W(k)^{1/k}\to\infty$, the example question, since the bound is exponential, which the OpenAI release's superexponential bound settles on its claim page; and the upper bound.

Depends on. No page of this wiki.

Standing. Claimed. The paper is an arXiv preprint with no journal record and no published independent review. The formal-conjectures file for the problem, at its commit of 2026-10-06, cites it as [CFS26] on the variant erdos_138.variants.dvd_two_pow ($W(k)/2^k\to\infty$), which it marks research solved with a Lean proof derived from Meta's Atlas proofs as its formal_proof; that Lean proof does not declare itself a formalization of this paper and has its own claim page. The OpenAI release manuscript of 23 September 2026 credits this paper with settling the $W(k)/2^k$ question. The site's commentary (page last edited 2 June 2026) does not mention the paper, and the site labels the problem OPEN. No Lean formalization of the paper's theorem is known, and this corpus has built nothing, so the page lists no evidence.