Classification of finite Euclidean Ramsey configurations

Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader–Russell–Walters conjecture that every such configuration is a subset of a finite transitive set. (Lean)

Manuscript

A classification of finite Euclidean Ramsey configurations

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026}

We classify finite Euclidean Ramsey configurations by a necessary and sufficient tensor condition over their coordinate fields. The Ramsey property here concerns monochromatic congruent copies at the original scale under arbitrary finite colorings. The criterion shows that every nonempty subtransitive set and every nonempty set of at most five points on a circle is Ramsey. In particular, some Ramsey cyclic quadrilaterals are not subtransitive, disproving the necessity direction of the Leader–Russell–Walters conjectured characterization.

Result 172 · Combinatorics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Agora members.