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Claim. I. Kříž, Permutation groups in Euclidean Ramsey theory, Proc. Amer. Math. Soc. 112 (1991), no. 3, 899--907, cited as [Kr91] in the site's commentary on Problem 174. Theorem 4.3 (p. 906) states that if $F$ is a finite Euclidean configuration and $G$ a soluble group of isometries of $F$, then for every $k\ge1$ there is a dimension $N$ such that every $k$-coloring of $\mathbb R^N$ has a congruent copy of $F$ on which every $G$-orbit is monochromatic; in particular, if some soluble group acts transitively on $F$, then $F$ and every set congruent to a subset of $F$ are Ramsey in the sense of Problem 174. Theorem 4.4 (p. 906) adds that a finite transitive configuration with a soluble group of isometries having at most two orbits is Ramsey. Corollary 4.5 (p. 907): the vertex set of every regular polygon is Ramsey. Corollary 4.6 (p. 907): the vertex sets of the regular tetrahedron, cube, octahedron, dodecahedron and icosahedron are Ramsey, the last two through a cyclic group of order ten with two orbits. The proof is an induction on the order of $G$ through a cyclic quotient, with a product theorem that refines colors on a finite witness and an orbit-gluing theorem that merges consecutive classes of an isometry orbit. The library's reconstructions are Theorem 4.3, Theorem 4.4, Corollary 4.5 and Corollary 4.6; the card is kriz_1991_permutation_groups_euclidean_ramsey_theory.

Covers. The class of subsoluble sets: every set congruent to a subset of a finite configuration with a transitive soluble group of isometries is Ramsey, as is every transitive configuration with a soluble group of at most two orbits; in particular all regular polygons and the five regular convex polyhedra. The paper does not decide sets outside these classes; the full classification is OpenAI's accepted claim.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper is a journal publication in the Proceedings of the American Mathematical Society, volume 112, issue 3 (July 1991), the refereed evidence; the issue carries no day, so this page is dated to the first day of that month. The site's curator credits regular polygons and polyhedra to [Kr91] in the problem's commentary, but the site labels the problem OPEN, so that credit is not reviewed evidence. The library's complete reconstructions, with their recorded source corrections, are the corpus's own reading and are not an independent review.