Shader 1976 all right triangles are ramsey e2

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corollary_4: Shader's corollary that every triangle with sides a, b and (b^2 + 2a^2)^{1/2} with 2b > a is Ramsey: every two-coloring of the plane contains a monochromatic triangle congruent to it.

corollary_5: Shader's corollary that every triangle with sides a, b and (4b^2 - a^2)^{1/2} with (3/2)^{1/2} b < a < (5/2)^{1/2} b is Ramsey: every two-coloring of the plane contains a monochromatic triangle congruent to it.

lemma_1: Shader's lemma that for any real number a and any two-coloring of the plane there is a monochromatic equilateral triangle of side ka for some k in {1, 3, 5, 7}, where k may depend on a.

theorem_2: Shader's theorem that every right triangle is Ramsey in the plane: every two-coloring of the plane contains a monochromatic triangle congruent to it.

theorem_3: Shader's theorem that for every parallelogram P and every two-coloring of the plane there is a parallelogram congruent to P with three vertices of one color.

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Leslie E. Shader, All Right Triangles Are Ramsey in E^2!. Journal of Combinatorial Theory, Series A 20 (1976), 385-389. doi:10.1016/0097-3165(76)90036-4. The file prints "Copyright © 1976 by Academic Press, Inc. All rights of reproduction in any form reserved." in its first-page footer, every other right reserved.

A triangle is called Ramsey in the plane if every two-coloring of the plane contains a monochromatic congruent copy. The paper proves four results (p. 385, labelled on pp. 388-389): every right triangle is Ramsey (Theorem 2); every parallelogram has a congruent copy in which three of the four vertices share a color (Theorem 3); each triangle with sides (a, b, sqrt(b^2 + 2a^2)) and 2b > a is Ramsey (Corollary 4); and each triangle with sides (a, b, sqrt(4b^2 - a^2)) and sqrt(3/2) b < a < sqrt(5/2) b is Ramsey (Corollary 5). The engine is Lemma 1 (p. 385): given a real a and a two-coloring of the plane, some equilateral triangle with side ka, for one of k = 1, 3, 5, 7 (k may depend on a), is monochromatic. By [2, Theorem 1] (Erdos) it suffices to find monochromatic copies of two triangles with odd integer sides (3, 5, 7 and 7, 15, 13) or of their odd multiples, and a case analysis on the colors of the points with integer coordinates in the frame spanned by an equilateral triangle of side 8 (Fig. 1) supplies them. Theorem 2 then follows by the "ladder" technique of [2], and the corollaries by applying Theorem 3 to a parallelogram and a rhombus. For problem 173 this is the statement-cited primary source for the right-triangle case, but it is only a special-case result and does not establish the conjecture, repeated here from earlier work [1] (Erdos, Graham, Montgomery, Rothschild, Spencer and Straus), that every non-equilateral triangle is Ramsey.

Source: https://doi.org/10.1016/0097-3165(76)90036-4.

Bears on. #173: Theorem 2 and Corollaries 4 and 5 show that no right triangle, no triangle with sides $a$, $b$, $\sqrt{b^2+2a^2}$ and $2b>a$, and no triangle with sides $a$, $b$, $\sqrt{4b^2-a^2}$ and $\sqrt{3/2}\,b<a<\sqrt{5/2}\,b$ can be the exceptional triangle of a two-coloring of the plane, and Lemma 1 that no two-coloring misses the equilateral triangles of all four sides $a$, $3a$, $5a$, $7a$. The paper says nothing about whether one coloring can miss two other triangles, which is the question.

Results. The printed statement of Corollary 5 gives the third side as $4b^2-a^2$, without the square root, while p. 385 and the proof give $(4b^2-a^2)^{1/2}$; on p. 385 the lower end of its range is printed without the factor $b$. The result pages record both.

• Lemma 1 (p. 385): monochromatic equilateral triangle of side $ka$, $k\in\{1,3,5,7\}$.
• Theorem 2 (p. 388): all right triangles are Ramsey.
• Theorem 3 (p. 388): every parallelogram has a congruent copy with three vertices of one color.
• Corollary 4 (p. 389): the triangles $(a,b,(b^2+2a^2)^{1/2})$, $2b>a$, are Ramsey.
• Corollary 5 (p. 389): the triangles $(a,b,(4b^2-a^2)^{1/2})$, $(3/2)^{1/2}\,b<a<(5/2)^{1/2}\,b$, are Ramsey.

Read status: claims checked for Lemma 1, Theorems 2 and 3 and Corollaries 4 and 5, read clause by clause on the page images of the print; the proofs read for structure only. The reduction and ladder technique of reference [2] are cited, not proved, in the paper and were not read. A second reader checked the result pages' statements, hypotheses, labels and pages against the print; the proofs were not independently reviewed.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.