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Claim. There is an absolute constant $c>0$ such that

$$W(3,k)\ge k^{c\log k/\log\log k}=\exp\Bigl(c\frac{(\log k)^2}{\log\log k}\Bigr),$$

equivalently $f(N)\le e^{C(\log N)^{1/2}(\log\log N)^{1/2}}$ for the least $k$ with $W(3,k)>N$, where $W(3,k)$ is the least $n$ such that every red/blue coloring of $\{1,\ldots,n\}$ has a red three-term or a blue $k$-term arithmetic progression (Hunter states it with the colors exchanged; the colors are names). This improves Green's bound $k^{c(\log k/\log\log k)^{1/3}}$ in the exponent. Hunter's Remark 1.1 records Green's expectation that $W(3,k)\le k^{O(\log k)}$ and infers that the bound is likely to be essentially best possible; no source states a matching lower bound as a conjecture. The theorem is paged at Theorem 1 of the library's source card; the locators are those of the arXiv v3 of 21 August 2022.

Covers. The lower-bound challenge of Problem 721, already met by Green on a separate claim page, with the best lower bound known as of the search. The upper-bound challenge is met by Schoen on Schoen's claim page, and the open-ended request for reasonable bounds is not covered: the order of magnitude is open between this bound and the site's $\exp(O((\log k)^9))$.

Depends on. Nothing in this wiki; the result is the paper's own theorem.

Acceptance. Reviewed: the site's curator, T. F. Bloom, labels the problem SOLVED and credits this improvement of the lower bound to the paper in the problem's commentary (page last edited 4 April 2026, accessed 2026-09-18). Refereed: Combinatorica 42 (2022), suppl. 2, 1231--1252 (the Crossref record, 2026-09-18); the locators are those of the arXiv preprint, whose v1 of 1 November 2021 is the first posting and names this page. Green's published article records the improvement in its June 2022 update note.

Read depth. Claims checked: Theorem 1, Remark 1.1 and footnote 1 (pp. 1--2 of the preprint); the proof was not read, the Combinatorica text was not compared with the preprint, and nothing is independently reviewed in this corpus.