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Statement

Setting (pp. 119--120). For a finite $X\subseteq\mathbb{E}^k$, $X$ is Ramsey if for every $r$ there is $N=N(X,r)$ such that every partition of $\mathbb{E}^N$ into $r$ classes has a class containing a congruent copy of $X$ (p. 119). $X$ is spherical if it lies on the surface of some sphere of finite radius (p. 120).

Conjecture (p. 120, unnumbered, with a prize, quoted). "If $X$ is spherical then $X$ is Ramsey."

The paper poses it after recording that the characterization of Ramsey sets "remains unanswered at the time of this writing" (p. 120), and after the known results it surveys there: every Ramsey set is spherical (Erdős et al., Euclidean Ramsey theorems I, Theorem 13), every simplex is Ramsey (Frankl and Rödl), Cartesian products of Ramsey sets are Ramsey, and a set with a solvable transitive automorphism group is Ramsey (Kříž), which covers the vertices of a regular pentagon. Together with the first of these, the conjecture would make the Ramsey sets exactly the spherical ones.

Read depth

Claims checked: the conjecture, the definitions it uses and its page were read on the print.

Dependencies

None in the corpus.

Source. R. L. Graham, Recent trends in Euclidean Ramsey theory, Discrete Math. 136 (1994), 119--127, doi:10.1016/0012-365X(94)00110-5; the edition read is named on the source card.

Bears on

• Problem 174: the problem asks for a characterization of the Ramsey sets. The paper records that question as unanswered in 1993 and conjectures that the spherical sets are the answer; it proves nothing toward the conjecture.