Erdos 1975 euclidean ramsey theorems iii

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conjecture_1: Conjectures that the only two-colorings of the plane with no monochromatic equilateral triangle of side d are colorings by alternate strips of width (sqrt(3)/2)d, up to some freedom on the strip boundaries.

conjecture_3: Conjecture 3 asserts that every non-equilateral triangle has a monochromatic congruent copy in every two-coloring of the plane; by Theorem 1 it is equivalent to Conjecture 2, that a coloring missing the equilateral triangle of one side has those of every other side.

corollary_10: States R(K) for every triangle in which the ratio of two sides is 2 sin(theta/2), with theta one of 30, 72, 108, 120 and 150 degrees.

corollary_20: States that in every proper two-coloring of the plane every right triangle whose acute angle alpha has alpha/90 degrees rational with even denominator occurs in all four possible colorings.

theorem_1: For a two-coloring f of the plane and a triangle K with sides a, b, c, states that f has a monochromatic congruent copy of K if and only if it has a monochromatic equilateral triangle of side a, b or c.

theorem_14: States that in every proper two-coloring of the plane a right triangle with legs a, b and b^2/a^2 rational occurs in all four possible colorings, and as Corollary 15 that R(a, b, c) holds whenever two sides are in ratio sqrt(2).

theorem_16: States that every proper two-coloring of the plane has a (1, 1, sqrt(3)) triangle with the 120 degree vertex colored opposite to the other two, so all three colorings of an isosceles 120 degree triangle occur.

theorem_17: States R(K) for every right triangle whose angle opposite one leg is a rational multiple of 180 degrees.

theorem_27: States that if R(1, 1, x) holds for some transcendental x < 2, then R(1, 1, y) holds for every y in some interval containing x in its interior.

theorem_28: States that the least planar set forcing a monochromatic (1, 1, x) triangle in every two-coloring has size tending to infinity as x tends to 1, and that the analogous bichromatic witness grows as x tends to 2.

theorem_5: For every two-coloring f of the plane, states that the set T_f of side triples (a, b, c) with no monochromatic triangle of those sides is totally disconnected in E^3.

theorem_6: For a two-coloring f and a right triangle with legs a, b, states that a monochromatic copy forces one with legs a/(2n+1), b, and that a copy with the b-side like-colored and the third point opposite forces a monochromatic copy with legs a/(2n), b and a like bichromatic one with legs a/n, b.

theorem_7: For a two-coloring f and right triangles K_alpha, K_beta with acute angles alpha, beta and equal hypotenuses, states that R_f(K_alpha) gives R_f(K_beta) when (2m+1)beta = alpha + n180 degrees, with two companion statements for bichromatic copies.

theorem_8: States, after R. M. Robinson, that if five planar points determine only the distances a, b, c, d, with d occurring once and a, b, c satisfying the triangle inequality, then every two-coloring of the plane has a monochromatic triangle with sides a, b, c.

theorem_9: States R(K) for every triangle with a 30 or 150 degree angle, for the triangles formed by the sides and circumradius of an isosceles triangle, and for the triangles satisfying any of four stated polynomial relations among the sides.

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Paul Erdős, Ronald L. Graham, Peter Montgomery, Bruce L. Rothschild, Joel Spencer, Ernst G. Straus, Euclidean Ramsey Theorems, III. Infinite and Finite Sets (Keszthely 1973), Colloquia Mathematica Societatis János Bolyai 10, North-Holland (1975), 559-583.

This third part restricts to n = 2, r = 2 and three-point sets K, asking for which triangles R(K) holds, i.e. every 2-coloring of the plane contains a monochromatic congruent copy. The organizing result is Theorem 1: for a triangle with sides a, b, c, R_f(K) holds for a coloring f if and only if R_f holds for at least one of the equilateral triangles of side a, b or c; Corollaries 2-4 turn this into transfer statements between isosceles and general triangles. Conjecture 1 asserts that the only colorings avoiding a monochromatic equilateral triangle of side d are the alternating strips of width (sqrt(3)/2)d; Conjecture 2 says a coloring avoiding side d has a monochromatic equilateral triangle of every other side d' != d; and Conjecture 3, equivalent to Conjecture 2 by Theorem 1, says R(K) holds for every non-equilateral triangle. Theorem 5 shows the exceptional set T_f is totally disconnected in E^3. Robinson's Theorem 8 (five points with only distances a, b, c, d where d occurs once) yields the seven families of Theorem 9 and Corollary 10, among them triangles with a 30- or 150-degree angle, triangles with a side ratio 2 sin(theta/2) for theta = 30, 72, 108, 120, 150 degrees, and degenerate (a, 2a, 3a) triples. The 'ladder' and 'roulette' methods (Theorems 6 and 7) and Theorem 14 (in every proper 2-coloring a right triangle with b^2/a^2 rational occurs in all four colorings; Corollary 15 adds side ratio sqrt(2)) yield right triangles: Theorem 17 proves R(K) for every right triangle with an angle a rational multiple of 180 degrees. Theorem 28 records the size barrier for finite witnesses: the minimal witness set S(x) for R(1,1,x) has |S(x)| tending to infinity as x tends to 1 (and likewise as x tends to 2 in the bichromatic case), proved by a limiting argument; the paper concludes that Conjectures 3 or 4 cannot be settled using finite subsets of bounded size. The paper poses problem #173 as its Conjecture 3.

Source: https://combinatorica.hu/~p_erdos/1975-12.pdf. No notice is printed; the file comes from the combinatorica.hu mirror of the Rényi Erdős archive, whose root could not be read, and the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, prints "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only."); the colloquium volume has no publisher page, and no Crossref license is recorded; the term is unstated.

Bears on.

• #173: Conjecture 3 (equivalently Conjecture 2) is the problem's statement, posed and not proved. Theorem 1 reduces it to whether a two-coloring can miss equilateral triangles of two different sides. Theorems 9, 14 and 17 and Corollaries 10 and 15 prove, for the triangles they name, a monochromatic congruent copy in every two-coloring, so none of those triangles is the exceptional triangle of any coloring; they do not settle the problem. Theorem 5 constrains the set of missed triangles, Theorem 27 is conditional and Theorem 28 limits finite methods; none decides a further triangle.

Results. Each page states the result with its printed label and page.

• Conjecture 1 (p. 560): the strip colorings are the only colorings missing a monochromatic equilateral triangle of side d.
• Conjectures 2 and 3 (p. 560): every non-equilateral triangle is Ramsey in the two-colored plane.
• Theorem 1 (p. 563), with the notation, Robinson's remark and Corollaries 2 to 4.
• Theorem 5 (p. 565): T_f is totally disconnected.
• Theorem 6 (p. 566): the ladder method.
• Theorem 7 (p. 568): the roulette method.
• Theorem 8 (p. 570): Robinson's five-point criterion.
• Theorem 9 (p. 572): seven families of Ramsey triangles.
• Corollary 10 (p. 573), with Corollary 11: side ratios 2 sin(theta/2).
• Theorem 14 (p. 574), with Theorem 12, Lemma 13 and Corollary 15.
• Theorem 16 (pp. 574--575): the isosceles 120-degree triangle in all three colorings.
• Theorem 17 (p. 576): right triangles with a rational angle.
• Corollary 20 (p. 577), with Theorems 18 and 19 and Conjecture 5.
• Theorem 27 (p. 582): a transcendental isosceles Ramsey triangle gives an interval.
• Theorem 28 (p. 583): minimal finite witnesses grow without bound.

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