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Claim. For every $\epsilon>0$ and every integer $t\ge1$ there is $N_0$ such that, whenever $N\ge N_0$, every $A\subseteq[t]^N$ with $|A|\ge\epsilon t^N$ contains a combinatorial line: a set $\{p_1,\ldots,p_t\}$ in which each coordinate is either constant or equal to $i$ on $p_i$, with at least one coordinate of the second kind. This is the density Hales--Jewett theorem of H. Furstenberg and Y. Katznelson, A density version of the Hales-Jewett theorem, J. Analyse Math. 57 (1991), 64--119, cited as [FuKa91] on the problem page, and it is exactly the question of Problem 171: the answer is yes. The paper is not held in the library; its statement is taken from the problem page and from Theorem 1.4 of the Polymath paper, which restates the theorem as Furstenberg and Katznelson's and proves it by a combinatorial density-increment argument in place of the original ergodic one (card). The Polymath reproof has its own claim page, Polymath 2012.

Depends on. No page of this wiki: the theorem is the paper's own, and the Polymath reproof is an independent second route, not an input.

Acceptance. Refereed: the paper appeared in Journal d'Analyse Mathématique, volume 57, issue 1; the publication record dates the issue to December 1991 without a day, so this page is named by the first day of that month. Reviewed: the site's curator, Thomas Bloom, records the problem as proved by Furstenberg and Katznelson in the commentary of the problem page (label PROVED (LEAN), page last edited 25 January 2026), and the Polymath paper's Theorem 1.4 attributes the theorem to them; that is documented acceptance outside this project. The site's Lean marker traces to a formalization of the Dodos--Kanellopoulos--Tyros proof of the theorem, pinned on their claim page, not of this one, so no formalized evidence is listed. Nothing here is this project's own review of the proof.