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Claim. G. Currier, K. Moore and C. H. Yip, Any two-coloring of the plane contains monochromatic 3-term arithmetic progressions, Combinatorica 44 (2024), no. 6, 1367--1380 (arXiv:2402.14197, v1 22 February 2024, v2 22 July 2024). Theorem 1.1 states that every two-coloring of $\mathbb R^2$ contains a monochromatic congruent copy of $\ell_3$, three collinear points with consecutive distance $1$, and hence, by scaling the coloring, a monochromatic three-term arithmetic progression of every common difference. Corollary 1.3 combines this with Theorem 1 of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus's 1975 paper (a two-coloring has a monochromatic $(a,b,c)$ triangle exactly when it has a monochromatic equilateral triangle of side $a$, $b$ or $c$): for $n\ge2$, every two-coloring of $\mathbb R^n$ has a monochromatic $(\alpha,2\alpha,x\alpha)$ triangle for every $\alpha>0$ and every $x\in[1,3]$. The proof of Theorem 1.1 goes through Lemma 2.1, whose first proof is a computer-checked finite gadget of fifty-six points, and Lemma 2.2 on the scaled hexagonal grid. The source is carded at currier_2024_any_two_coloring_plane_contains_monochromatic.

Covers. The statement of Problem 173 for the degenerate triangle of three equally spaced collinear points, at every scale, and for every triangle with sides $(\alpha,2\alpha,x\alpha)$, $1\le x\le3$: none of them is the exceptional triangle of any two-coloring of the plane. The paper settles no other triangle and says nothing about whether one coloring can miss two triangles.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper is a journal publication in Combinatorica, volume 44, issue 6 (2024), the refereed evidence; this page is dated to the first arXiv posting, the journal record carrying no day known here. The site's commentary does not cite the paper; the problem's discussion on the site cites it (15 February 2026), which is commentary on a problem the site labels OPEN and not reviewed evidence. The proof is not checked by this corpus, and nothing is independently reviewed by this project.