Cantwell 1996 finite euclidean ramsey theory

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theorem_2_11: Cantwell's theorem that every 2-coloring of four-dimensional Euclidean space contains a monochromatic square of the prescribed side b, two dimensions below the theorem of Erdős et al. for six-dimensional space.

theorem_3_1: Cantwell's theorem that the chromatic number of four-dimensional Euclidean space is at least 7, raising the lower bound of 6 that the paper attributes to Larman and Rogers.

theorem_4_1: Cantwell's theorem that the chromatic number of five-dimensional Euclidean space is at least 9, raising the lower bound of 8 that the paper records as Theorem 4.2.

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Kent Cantwell, Finite Euclidean Ramsey Theory. Journal of Combinatorial Theory, Series A 73 (1996), 273–285. doi:10.1016/S0097-3165(96)80006-9. The copy read prints "Copyright © 1996 by Academic Press, Inc. All rights of reproduction in any form reserved." (p. 273), every other right reserved.

Theorem 2.11 (p. 278; the introduction on p. 273 cites it as Theorem 2.10) proves that any 2-coloring of R^4 contains a monochromatic square of any prescribed side b, lowering the dimension in the earlier Erdős et al. result for R^6 (Theorem 2.1, p. 274) by two, and by one from its later form in R^5 (which holds because the fifteen relevant points lie in a hyperplane). Theorem 3.1 (p. 279) uses that square theorem; Theorem 3.1 and Theorem 4.1 (p. 281) raise the chromatic number lower bounds for R^4 and R^5 from 6 and 8 to 7 and 9 respectively, equivalently establishing R(K,4,6) and R(K,5,8) for K a unit-distance pair. The method is geometric and finitary: the standard configuration of side b (the points of R^5 with two coordinates equal to b/sqrt(2) and the rest 0, which lie in a hyperplane) is used to translate colorings of space into edge-colorings of the complete graph on 5 points, so that 4-cycles, 3-cycles, stars of four edges and complete subgraphs on 4 points correspond to squares, equilateral triangles, regular tetrahedra and octahedra of side b (Lemma 2.2, pp. 274–275). The square theorem then counts monochromatic equilateral triangles over a grid of four-dimensional cross polytopes; the chromatic bounds proceed by case analysis on such configurations (Section 3) and on the half cube in R^5 (Section 4). The paper proves statements about all 2-colorings of R^4 and about the chromatic numbers of R^4 and R^5; it says nothing about the plane.

Read status: claims checked for Theorems 2.11, 3.1 and 4.1, the definition of the chromatic number and Lemma 2.2, read clause by clause on the page images of the print; the proofs read for structure only. Nothing here is independently reviewed.

Source: https://doi.org/10.1016/S0097-3165(96)80006-9.

Bears on. #214: the problem asks whether, in the plane, the complement of a set with no two points at distance 1 must contain a unit square. Theorem 2.11 gives a monochromatic square of side b in every 2-coloring of R^4, with no distance hypothesis on either color; it asserts nothing about the plane, and the paper does not raise the problem's question.

Results.

• Theorem 2.11 (p. 278): for every b > 0, any 2-coloring of R^4 yields a monochromatic square of side b.
• Theorem 3.1 (p. 279): the chromatic number of R^4 is at least 7, improving the previous lower bound of 6.
• Theorem 4.1 (p. 281): the chromatic number of R^5 is at least 9, improving the previous lower bound of 8.
• Lemma 2.2 (pp. 274–275), no page of its own: the map sending edge ij of the complete graph on 5 points to the point of the standard configuration of side b with nonzero coordinates i and j sends 4-cycles to squares, 3-cycles and stars of three edges to equilateral triangles, stars of four edges to regular tetrahedra, complete subgraphs on 4 points to octahedra, and those minus one edge to square-based pyramids, all of side b.

The copy read for this card is the journal's PDF of that edition.

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