Graham 2017 euclidean ramsey theory

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conjecture_11_1_1: Graham's survey restates the conjecture that for every nonequilateral triangle T, every partition of the plane into two classes has a class containing a congruent copy of T.

conjecture_11_1_2: Graham's survey conjectures that for every partition of the plane into two classes, one of the classes contains a congruent copy of every triangle, with the possible exception of a single equilateral triangle.

conjecture_11_1_3: Graham's survey conjectures that for every triangle T there is a partition of the plane into three classes none of which contains a congruent copy of T.

conjecture_11_2_13: Graham's conjecture, carrying a prize, that every spherical set is Ramsey, which with Theorem 11.2.5 would make the Ramsey sets exactly the spherical sets.

conjecture_11_2_14: Graham's survey restates the conjecture of Leader, Russell and Walters that every Ramsey set is a subset of a finite configuration with a transitive group of symmetries.

conjecture_11_5_6: Graham's survey restates Erdős's conjecture that a set of positive integers whose reciprocals have divergent sum contains arbitrarily long arithmetic progressions.

conjecture_11_6_5: Graham's survey records the Erdős-Szekeres theorem that a least f(n) exists with every f(n) points in general position in the plane containing a convex n-gon, the bounds known in 2017, and the conjecture that f(n) = 2^(n-2)+1 for n >= 3.

problem_11_1_6: Graham's survey defines the chromatic number of n-space through the unit distance pair, records the bounds 4 <= chi(E^2) <= 7 as standing in 2017, and poses the problem of determining chi(E^2) exactly.

theorem_11_1_4: Graham's survey collects which triangles are known to be 2-Ramsey in the plane (nine families), that every nondegenerate triangle is 2-Ramsey in 3-space, and known positive and negative cases for right triangles, squares, rectangles and degenerate triangles.

theorem_11_2_5: Graham's survey records that every Ramsey set lies on the surface of some sphere, the necessary condition set against the sufficient conditions of Section 11.2.

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R. L. Graham, Euclidean Ramsey Theory. Handbook of Discrete and Computational Geometry, 3rd edition, Chapter 11, CRC Press (preliminary version, August 10, 2017). The copy read for this card is the author's preliminary version, whose page footers read only "Preliminary version (August 10, 2017). To appear in the Handbook of Discrete and Computational Geometry, ... 3rd edition, CRC Press, Boca Raton, FL, 2017"; its hosting page states no copyright, permission or terms (https://www.csun.edu/~ctoth/Handbook/HDCG3.html, read 2026-10-02), and the publisher's page describes the printed edition, not that preliminary version, so it was not consulted; the term is unstated.

The chapter is a survey of Euclidean Ramsey theory: for a finite set X in E^N, E^N r-> X means that every partition of E^N into r classes has a class containing a congruent copy of X, and X is Ramsey when this holds for every r once N is large enough (p. 281). Section 11.1 opens with three plane conjectures: Conjecture 11.1.1, that every nonequilateral triangle is 2-Ramsey in the plane; the stronger Conjecture 11.1.2, that in every two-coloring of the plane one class contains a congruent copy of every triangle, one equilateral triangle perhaps excepted; and Conjecture 11.1.3, that no triangle is 3-Ramsey in the plane (p. 282). The equilateral exception is needed, since the coloring by alternating half-open strips of width 1 has no monochromatic equilateral triangle of side sqrt(3); the chapter also notes the zebra-like colorings of Jelinek, Kyncl, Stolar and Valla, and their result that when the plane is split into an open set and a closed set, every equilateral triangle occurs in at least one of the two (p. 282). Theorem 11.1.4 (pp. 282-283) collects the known cases, reported from Euclidean Ramsey Theorems III and later papers: nine families of triangles 2-Ramsey in the plane (among them every triangle with a 30, 90 or 150 degree angle, and the degenerate (a,2a,3a)), every nondegenerate triangle 2-Ramsey in E^3, every nondegenerate right triangle 3-Ramsey in E^3, and every rectangle 2-Ramsey in E^5, against negative items for the (30,60,90) triangle in E^3 with 12 colors, the square in E^2 and E^4 with 2 colors, and the degenerate (1,1,2) and (a,b,a+b) triangles in every dimension with 4 and 16 colors. Section 11.1 continues with the chromatic number of E^n, recording 4 <= chi(E^2) <= 7 and posing Problem 11.1.6, to determine chi(E^2) (pp. 283-284).

Section 11.2 (pp. 284-286) surveys Ramsey sets: products of Ramsey sets, rectangular sets, simplices (Frankl and Rodl) and sets with a transitive solvable group of isometries (Kriz) are Ramsey, while Theorem 11.2.5 states that every Ramsey set is spherical. Near its end it states Conjecture 11.2.12 on 4-point subsets of a circle, Graham's Conjecture 11.2.13 that every spherical set is Ramsey, and the rival Conjecture 11.2.14 of Leader, Russell and Walters that every Ramsey set is subtransitive. Later sections cover sphere-Ramsey sets (11.3), edge-Ramsey sets (11.4), homothetic Ramsey sets and density theorems (11.5), including van der Waerden numbers and Erdos's Conjecture 11.5.6 on sets with divergent reciprocal sum, and variations (11.6): asymmetric results such as Juhasz's E^2 2-> (P_2, T_4) for every four-point T_4 and the eight-point T_8 of Csizmadia and Toth with E^2 not 2-> (P_2, T_8), partitions into infinitely many parts, and the Erdos-Szekeres theorem with Conjecture 11.6.5 (pp. 292-294). The chapter proves none of its numbered results in full; they are reported from the sources it cites.

Source: https://www.csun.edu/~ctoth/Handbook/chap11.pdf.

Read status. Claims checked: the statements linked below were read clause by clause on the printed pages of the preliminary version. The chapter gives no proofs of them, and none was checked here.

Bears on.

• #173: Conjecture 11.1.2 implies the problem's statement; Conjecture 11.1.1 asserts, for each nonequilateral triangle separately, that no two-coloring of the plane misses it, and is not a restatement of the problem; Theorem 11.1.4(a) reports triangle families that no two-coloring of the plane misses. The chapter settles the problem for no coloring.
• #174: Theorem 11.2.5 gives the necessary condition that Ramsey sets are spherical; Conjecture 11.2.13 proposes that the Ramsey sets are exactly the spherical sets, and Conjecture 11.2.14 the necessary condition that Ramsey sets are subtransitive. The chapter characterizes no class of sets as exactly the Ramsey sets.
• #508: Problem 11.1.6 is the problem's question; the chapter records the bounds 4 <= chi(E^2) <= 7 as of 2017.
• #3: Conjecture 11.5.6 restates the problem's question as a conjecture of Erdos.
• #107: Conjecture 11.6.5 is the problem's equality f(n) = 2^(n-2) + 1, posed for n >= 3 and points in general position; the chapter records the bounds 2^(n-2) + 1 <= f(n) <= 2^(n + 4n^(4/5)), the upper one for large n.

Results. Conjecture 11.1.1 (p. 282); Conjecture 11.1.2 (p. 282); Conjecture 11.1.3 (p. 282); Theorem 11.1.4 (pp. 282-283); Problem 11.1.6 (p. 284, with the definition and bounds on p. 283); Theorem 11.2.5 (p. 285); Conjecture 11.2.13 (p. 286, with Conjecture 11.2.12); Conjecture 11.2.14 (p. 286); Conjecture 11.5.6 (p. 291, with Conjecture 11.5.7); Conjecture 11.6.5 (p. 294, with Theorem 11.6.4).

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