---

Claim. There are absolute constants $C,c>0$ such that for every $k$

$$W(3,k)\le\exp\bigl(Ck^{1-c}\bigr),$$

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 (Schoen states it as a partition of $\{1,\ldots,N\}$ into a set without three-term progressions and a set without $k$-term progressions; the colors are names). This is the second explicit challenge of Problem 721, to prove $W(3,k)<\exp(k^c)$ for some $c<1$, with Schoen's exponent $1-c$ in the role of the problem's $c$. The theorem is paged at Theorem 1 of the library's source card. The proof adapts the author's method on the structure of large spectra from Schoen's Adv. Math. 2021 paper on Roth's theorem to use the structure of both partition classes. The paper's remark after the theorem notes that the 2020 Bloom--Sisask bound $N/(\log N)^{1+c}$ for sets without three-term progressions already implies the same shape with a far smaller $c$.

Covers. The upper-bound challenge: $W(3,k)<\exp(k^c)$ for some constant $c<1$. It does not cover the lower-bound challenge, met by Green and Hunter on their own claim pages, nor the open-ended request for reasonable bounds, since the true order of magnitude of $W(3,k)$ is open between $\exp(c(\log k)^2/\log\log k)$ 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 Schoen in the problem's commentary with the first bound of the shape $\exp(k^c)$, $c<1$ (page last edited 4 April 2026, accessed 2026-09-18). Refereed: Electron. J. Combin. 28 (2021), no. 2, P2.34, 10 pp., submitted 9 July 2020, accepted 23 May 2021 and published 4 June 2021 (the article's first page). The arXiv preprint 2006.02877v1 of 4 June 2020 is the first posting and names this page. Semantic Scholar's three citing records, include no dispute.

Read depth. Claims checked: Theorem 1 and the remark after it (p. 2); Section 3 (pp. 4--9) for structure only. Nothing is independently reviewed in this corpus.