Nearly all known Euclidean Ramsey sets are subsoluble

TL;DR

This work investigates the relationship between Euclidean Ramsey sets and subsolubility, building on Kříž's soluble-group criterion for Ramsey sets. It shows that nearly all known Ramsey sets can be embedded into soluble sets (i.e., are subsoluble) by developing results for block-set permutations, regular polytopes, and isosceles trapezia, often via higher-dimensional embeddings and carefully constructed soluble group actions. The paper also identifies two potential exceptions (the 120-cell and 600-cell) and raises open questions about whether every Ramsey set must be subsoluble or embed into a soluble set, as well as the broader implications for block-set conjectures and transitivity constraints. Overall, the results suggest a pervasive link between Ramsey properties and soluble symmetry structures in Euclidean spaces, with important implications for rival conjectures in the field.

Abstract

A finite set $X$ in a Euclidean space $\mathbb{R}^d$ is called Ramsey if for every $k$ there exists an integer $n$ such that whenever $\mathbb{R}^n$ is coloured with $k$ colours, there is a monochromatic copy of $X$. Graham conjectured that all spherical sets are Ramsey, but progress on this conjecture has been slow. A key result of Kříž is that all sets that embed in sets that are acted on transitively by a soluble group are Ramsey. We show that for nearly all known examples of Ramsey sets the converse is true, with only two possible exceptions.

Figures (3)

• Figure 1: A face-coloured icosahedron with pyritohedral symmetry.
• Figure 2: The cube inscribed in the dodecahedron.
• Figure 3: Implications between properties of finite sets in Euclidean space

Theorems & Definitions (19)

• Conjecture 1.1: graham
• Theorem 1.2: kriz
• Definition 1
• Theorem 1.3: Karamanlis
• Conjecture 1.4: blocksets
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 3.1