Szlam 2001 monochromatic translates configurations plane

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proposition_2: Szlam's partial converse to his reduction, turning a proper n-coloring of R^m whose color classes are translates of one class into an admissible red-blue coloring with no red translate of some n-point configuration.

theorem_1: Szlam's theorem that every red-blue coloring of the plane with no two blue points at distance one contains a red translate of every three-point configuration, with an analogue in R^m for n-point configurations whose printed range n <= (1+o(1))(1.2)^n has a misprinted exponent.

theorem_2: Szlam's theorem that some red-blue coloring of the plane with no two blue points at distance one avoids every red translate of some seven-point configuration.

theorem_3: Szlam's dichotomy that either every admissible red-blue coloring of the plane has a red translate of every four-point configuration, or some admissible coloring forbids red congruent copies of some seven-point configuration.

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Arthur D. Szlam, Monochromatic Translates of Configurations in the Plane. Journal of Combinatorial Theory, Series A 93 (2001), 173--176. doi:10.1006/jcta.2000.3065.

Call a red-blue coloring admissible if no two blue points are at distance one. Theorem 1 (p. 174) shows that in every admissible coloring of the plane each three-point configuration has an all-red translate, and more generally that in every admissible coloring of R^m each n-point configuration has an all-red translate for n up to (1 + o(1))(1.2)^n as printed; the proof derives this from the Frankl-Wilson bound on chi(R^m), which is exponential in m, so the range reads with exponent m, a correction the paper does not state. Theorem 2 (p. 174) gives a seven-point configuration and an admissible coloring of the plane under which no translate of that configuration is all red. The mechanism is a two-way reduction: Proposition 1 (p. 174) turns an admissible coloring avoiding red translates of an n-point set into a proper n-coloring of R^m (each color class avoiding distance one), so chromatic-number lower bounds give Theorem 1, while Proposition 2 (p. 174) converts a regular proper n-coloring (one whose color classes are translates of the first) back into such a coloring, and Isbell's hexagonal coloring yields Theorem 2. Theorem 3 (p. 175) links translates to congruent copies: either every admissible coloring of the plane has a red translate of every four-point configuration, or some admissible coloring forbids red congruent copies of some seven-point configuration, a Moser spindle in the proof. The paper recalls (p. 173) Juhász's theorem that every admissible coloring of the plane has a red congruent copy of every four-point configuration.

Source: https://doi.org/10.1006/jcta.2000.3065. The print carries "Copyright © 2001 by Academic Press" and "All rights of reproduction in any form reserved." on its first page.

Bears on. #214: the problem asks for a unit square in the complement of a set avoiding distance one, which is the red set of an admissible coloring. Theorem 1 gives red translates, hence red congruent copies, of every three-point configuration only, and does not reach four points. Theorem 2 and Proposition 2 forbid red translates, not red congruent copies, and Theorem 3 is a dichotomy the paper does not resolve; none of them decides the unit-square question or changes the bounds on the largest size of configuration whose congruent copies are forced in the red set.

Results.

• Theorem 1 (p. 174): every admissible coloring of the plane has a red translate of every three-point configuration, with the analogue in R^m and Proposition 1 (p. 174), the reduction to a proper n-coloring.
• Theorem 2 (p. 174): some admissible coloring of the plane has no all-red translate of some seven-point configuration.
• Theorem 3 (p. 175): either every admissible coloring of the plane has a red translate of every four-point configuration, or some admissible coloring has no red congruent copy of some seven-point configuration.
• Proposition 2 (p. 174): a regular proper n-coloring of R^m yields an admissible two-coloring and an n-point configuration with no red translate.

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