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Updated Oct 8

Problem 173

claims/: The 3 claim pages of Problem 173, one per claimant's result; the problem's standing derives from them.

Statement. In any $2$ -colouring of $\mathbb{R}^2$ , for all but at most one triangle $T$ , there is a monochromatic congruent copy of $T$ .

Status. Open, the site's label (page last edited 16 October 2025). No result settles the question. Three results decide the question for particular triangles and are recorded as partial claims: Shader's refereed theorem of 1976 that every right triangle, and every triangle of two further one-parameter families, has a monochromatic congruent copy in every two-coloring of the plane, on Shader 1976 (accepted on the refereed publication alone); the triangle families of the 1975 colloquium paper of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, on EGMRSS 1975 (claimed, a proceedings volume); and Currier, Moore and Yip's refereed theorem of 2024 that three equally spaced collinear points, and every $(\alpha,2\alpha,x\alpha)$ triangle with $1\le x\le3$ , appear monochromatically, on Currier, Moore and Yip 2024 (accepted on the refereed publication alone). No full claim exists, and the standing derives from the claim pages.

Source. erdosproblems.com/173, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #173, https://www.erdosproblems.com/173.

References.

[Sh76] Shader, L., All right triangles are Ramsey in $\mathbb{E}^2$ ! J. Comb. Th. A 20 (1976), 385-389.

Formalization. None recorded.

Current assessment

The question (site formulation, page last edited 16 October 2025). The statement above; OPEN. The commentary, in this page's words: some colorings force one equilateral triangle to be excluded, the coloring of the plane by alternating strips being the example, and Shader [Sh76] proved the statement for any single right triangle. The 1975 colloquium paper of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus poses the question as its Conjecture 3, that every non-equilateral triangle has a monochromatic congruent copy in every two-coloring of the plane, and shows by its Theorem 1 that a coloring misses a triangle with sides $a$ , $b$ , $c$ exactly when it misses the equilateral triangles of all three side lengths; so the question is whether a two-coloring of the plane can miss equilateral triangles of two different sides. The discussion on the site held one comment (15 February 2026) citing that paper, Graham's 2017 survey of Euclidean Ramsey theory, Currier, Moore and Yip's 2024 paper and the 2025 computational paper of Mundinger and coauthors, and the proof-claim tab was empty.

Settled triangles. The three claim pages named under Status record the triangles for which the question is decided: Shader's right triangles, the triangles with sides $a$ , $b$ , $\sqrt{b^2+2a^2}$ , $2b>a$ , and the triangles with sides $a$ , $b$ , $\sqrt{4b^2-a^2}$ , $\sqrt{3/2}\,b<a<\sqrt{5/2}\,b$ (refereed, accepted); the families of the 1975 paper (triangles with a side ratio $2\sin(\theta/2)$ for $\theta$ equal to $30$ , $72$ , $90$ or $120$ degrees, triangles with an angle of $30$ or $150$ degrees, the degenerate triples $(a,2a,3a)$ and right triangles with $b^2/a^2$ rational; a proceedings volume, claimed); and the degenerate triangle $\ell_3$ of three equally spaced collinear points with the $(\alpha,2\alpha,x\alpha)$ triangles, $1\le x\le3$ , of Currier, Moore and Yip (refereed, accepted). Each decides that the triangles it names are not the exceptional triangle of any coloring; none bears on whether one coloring can miss two other triangles.

Results without a claim page. Jelínek, Kynčl, Stolař and Valla, Combinatorica 29 (2009), 699--718, prove the statement under restricted colorings. For partitions of the plane into a closed and an open set they prove it for every triangle (Theorem 2.1). For polygonal colorings, whose avoiding colorings they classify as the zebra-like ones, they prove it for every non-equilateral triangle, with equilateral triangles of at most one side missed (Theorem 3.20). Those results restrict the colorings, so they decide no triangle for all two-colorings and settle no instance of the question; they rule out the natural candidate counterexamples. Frankl and Rödl's 1986 theorem that all triangles are Ramsey concerns high dimensions and every number of colors and settles no instance of the two-dimensional two-color question. The 1975 paper's Theorem 28 shows that the least planar witness set for the triangle $(1,1,x)$ grows without bound as $x\to1$ , a limit on finite methods rather than a result on the question. Graham's survey and the computational search of Mundinger and coauthors (2025) prove no new triangle Ramsey. The library cards linked below record the remaining sources.

Remaining gaps. Whether a two-coloring of the plane can miss equilateral triangles of two different sides is open, and with it the question. Proof coverage is at statement level throughout: no proof of the claim pages' theorems is checked by this corpus, and nothing is independently reviewed by this project.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (77)

The triangle families of Euclidean Ramsey Theorems III Shader's right triangles and two algebraic triangle families Currier, Moore and Yip's monochromatic three-term progressions problems/discrete_geometry/E0173/claims Discrete and Convex Geometry additive_combinatorics/erdos_1979_old_new_problems_results_combinatorial_number Discrete and Convex Geometry discrete_geometry/aichholzer_2019_triangles_colored_euclidean_plane Theorem 3.2 (p. 10:6): k-color zebra colorings avoid non-obtuse normed triangles with BC at least 1/(k-1)discrete_geometry/bialostocki_2006_minimum_sets_forcing_monochromatic_triangles discrete_geometry/currier_2024_any_two_coloring_plane_contains_monochromatic Corollary 1.3 (p. 2): every (alpha, 2 alpha, x alpha) triangle with x in [1,3] is two-color Ramsey in E^n, n >= 2 Lemma 2.1 (p. 3): with no monochromatic unit-step progression, a red-blue-blue unit triangle has a blue centroid Lemma 2.2 (p. 5): with no monochromatic unit-step progression, the 1/sqrt(3) hexagonal grid has one coloring up to isometry Theorem 1.1 (p. 1): every two-coloring of the plane has a monochromatic unit-step three-term progression Euclidean Ramsey Theorems I Conjecture, p. 347: every non-equilateral triangle is forced in two-colorings of the plane Corollary 10 — the 30°–60° right triangle in the plane Euclidean Ramsey I — historical questions and later links Theorem 9 — one of three prescribed planar triangles discrete_geometry/erdos_1975_euclidean_ramsey_theorems_iii Conjecture 1 (p. 560): the strip colorings are the only ones missing an equilateral triangle Conjectures 2 and 3 (p. 560): every non-equilateral triangle is Ramsey in the two-colored plane Corollary 10 (p. 573): triangles with a side ratio 2 sin(theta/2)Corollary 20 (p. 577): right triangles with a rational angle of even denominator occur in all four colorings Theorem 1 (p. 563): a triangle is monochromatic exactly when an equilateral triangle on one of its sides is Theorem 14 (p. 574): right triangles with b^2/a^2 rational occur in all four colorings Theorem 16 (p. 574): all three colorings of the isosceles 120-degree triangle occur Theorem 17 (p. 576): right triangles with a rational angle are Ramsey in the two-colored plane Theorem 27 (p. 582): a transcendental isosceles Ramsey triangle would give an interval of them Theorem 28 (p. 583): minimal finite witnesses for (1, 1, x) grow without bound as x tends to 1 Theorem 5 (p. 565): the set of missed triangles is totally disconnected Theorem 6 (p. 566): the ladder method for right triangles Theorem 7 (p. 568): the roulette method for right triangles Theorem 8 (p. 570): Robinson's five-point criterion Theorem 9 (p. 572): seven families of triangles that are Ramsey in the two-colored plane discrete_geometry/frankl_1986_all_triangles_are_ramsey Theorem 1 (p. 777): every triangle is Ramsey discrete_geometry/graham_2017_euclidean_ramsey_theory Conjecture 11.1.1 (p. 282): every nonequilateral triangle is 2-Ramsey in the plane Conjecture 11.1.2 (p. 282): one class of a two-coloring of the plane holds every triangle but one equilateral Conjecture 11.1.3 (p. 282): no triangle is 3-Ramsey in the plane Theorem 11.1.4 (pp. 282-283): the known triangle and quadrilateral cases of low-dimensional r-Ramsey questions discrete_geometry/grytczuk_2016_fractional_j_fold_colouring_plane Theorem 1 (p. 5, quoted from Nielsen): a monochromatic limit triangle congruent to any triangle in a two-colouring of the plane discrete_geometry/jelinek_2009_monochromatic_triangles_two_colored_plane Corollary 1.4 (p. 4): containing all triangles reduces to containing equilateral ones Theorem 2.1 (p. 4): a coloring by a closed and an open set contains every triangle Theorem 3.19 (p. 18): every zebra-like coloring has a twin avoiding the unit triangle Theorem 3.20 (p. 18): polygonal colorings contain every non-equilateral triangle off the boundary

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