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Claim. Theorem 1.1 of Kelley and Meka, Strong Bounds for 3-Progressions, states that there is an absolute constant $\beta>0$ such that every $A\subseteq\{1,\ldots,N\}$ with no non-trivial three-term arithmetic progression has density at most $2^{-\Omega((\log N)^\beta)}$, that is,

$$r_3(N)\le N\,2^{-c(\log N)^{\beta}}$$

for some $c>0$ and all large $N$, where $r_3(N)$ is the largest size of such a set. The quantitative form is Theorem 1.2: a set of density at least $2^{-d}$ has at least $2^{-O(d^{12})}N^2$ solutions of $x+y=2z$, so $\beta$ can be taken as $1/12$. The result page Theorem 1.1 records both statements. Since the factor $2^{-c(\log N)^\beta}$ tends to zero, the bound gives $r_3(N)=o(N)$, the instance $k=3$ of Problem 139, first proved by Roth in 1953, with a rate the problem does not ask for. The same theorem is the accepted full claim of Problem 140, recorded on its claim page there.

Covers. The instance $k=3$ of the statement, $r_3(N)=o(N)$, which Szemerédi's accepted full claim already settles; the page records the bound's rate, which no claim of this problem requires. Nothing about any $k\ge4$.

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

Acceptance. Reviewed: Bloom and Sisask, two named experts on the problem, re-derived the full argument in their refereed exposition The Kelley--Meka bounds for sets free of three-term arithmetic progressions, Essential Number Theory 2 (2023), no. 1, 15--44, doi:10.2140/ent.2023.2.15, and then sharpened the exponent to $1/9$ in arXiv:2309.02353, which has its own claim page. The site's curator labels the problem proved on Szemerédi's theorem and cites this paper in the commentary only as the best known bound for $k=3$, which credits the bound and not a settlement of this problem; the curator's acceptance of the same theorem as the proof of Problem 140 is recorded on that problem's claim page. Not refereed: the paper appeared in the proceedings of the 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS 2023), pp. 933--973, doi:10.1109/FOCS57990.2023.00059, a conference proceedings and not a journal, and no journal version is recorded (Crossref, 2026-09-18), so refereed is not listed. The proof is not checked here.