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Statement

Definitions as on the Theorem 11.2.5 page: a finite set is Ramsey when, for every number of colors, every coloring of a Euclidean space of large enough dimension has a monochromatic congruent copy of it, and spherical when it lies on the surface of some sphere.

Conjecture 11.2.13 (p. 286), with a prize printed beside the label. Every spherical set is Ramsey.

The chapter calls determining the Ramsey sets "the outstanding open problem in Euclidean Ramsey theory" and observes that the conjecture, if true, would make the Ramsey sets exactly the spherical sets (by Theorem 11.2.5). It places before it Conjecture 11.2.12 (p. 286), that every 4-point subset of a circle is Ramsey, with Křiž's proof of the case where two opposite sides are parallel.

Scope

This is a conjecture, not a result proved in the chapter. The rival Conjecture 11.2.14 of Leader, Russell and Walters follows it on the same page.

Source. R. L. Graham, Euclidean Ramsey theory, Chapter 11 of J. E. Goodman, J. O'Rourke and C. D. Tóth (eds.), Handbook of Discrete and Computational Geometry, 3rd edition, CRC Press, Boca Raton, FL, 2017; Conjectures 11.2.12 and 11.2.13 and the remarks around them on p. 286. Pages are those printed on the edition named on the source card.

Read depth. Claims checked: the statement and the sentences around it were read on the printed page.

Bears on

• Problem 174: the conjecture proposes a characterization of the Ramsey sets, the spherical sets, which is the kind of answer the problem asks for; the chapter states only the necessity half (Theorem 11.2.5), without proof. The problem page records the later work that bears on this conjecture.