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Claim. Theorem 1.1 of Green and Tao, New bounds for Szemerédi's theorem, III: a polylogarithmic bound for $r_4(N)$, states that there is an absolute constant $c>0$ with

$$r_4(N)\ll N(\log N)^{-c},$$

where $r_4(N)$ is the largest size of a subset of $\{1,\ldots,N\}$ with no non-trivial four-term arithmetic progression. Since $(\log N)^{-c}\to0$, the bound gives $r_4(N)=o(N)$, the instance $k=4$ of Problem 139, together with a rate the problem does not ask for. The paper improves Gowers's bound $N(\log\log N)^{-c}$ and the authors' earlier $N\exp(-c\sqrt{\log\log N})$, and brings the four-term bound to the quality of the Heath-Brown–Szemerédi bound for $r_3(N)$; its method replaces Roth's density increment by an energy decrement over Bohr sets with a local inverse theorem for the $U^3$ norm. The library card is Green and Tao 2017.

Covers. The instance $k=4$ of the statement, $r_4(N)=o(N)$, which Szemerédi's accepted full claim already settles; the page records the bound's rate, $N(\log N)^{-c}$, which no claim of this problem requires. Nothing about any other $k$.

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

Acceptance. Refereed: B. Green and T. Tao, New bounds for Szemerédi's theorem, III: a polylogarithmic bound for $r_4(N)$, Mathematika 63 (2017), no. 3, 944–1040, the DOI linked above; the page name carries the date of the first arXiv version, 2017-05-04. Not reviewed: 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 $k=4$, which credits the bound and not a settlement of the problem, so no reviewed evidence is listed. The proof is not checked here.