Result 172, Combinatorics

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 formalization Classification or exact value

The bigger picture

Why it matters

Can a finite pattern of points be forced to appear in one color, no matter how space is colored? The manuscript reports an algebraic classification of these unavoidable patterns, while showing that symmetry does not explain all of them.

What changes?

A finite configuration is Euclidean Ramsey if, for every finite number of colors, all colorings of sufficiently high-dimensional Euclidean space contain a monochromatic congruent copy. The copy keeps the original distances: resizing is not allowed. The required dimension can depend on the configuration and the number of colors. The manuscript claims a necessary and sufficient tensor condition over the coordinate field, the field generated by the point coordinates, for this property.

What does that help mathematicians do?

The criterion reportedly establishes that every nonempty subset of a finite transitive set is Ramsey. Here, transitive means that symmetries can move any point of the set to any other. It also covers every nonempty set of at most five points on a circle. Crucially, some Ramsey quadrilaterals on circles cannot sit inside any finite transitive set. This disproves the necessity direction of the Leader-Russell-Walters conjecture: being contained in such a symmetric set is sufficient, but not necessary.

Are there practical applications?

The immediate value is foundational: the claimed classification connects an unavoidable-color-pattern question with algebraic structure in the coordinates. Researchers gain a framework for studying which fixed-distance patterns must recur and for separating symmetry-based explanations from more general ones. The supplied material does not establish a practical computational method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A classification of finite Euclidean Ramsey configurations

September 23, 2026 26 pages Main result formalized in Lean

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.

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Cite (BibTeX) 
@misc{OAI:A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026,
  author = {{OpenAI}},
  title = {{A classification of finite Euclidean Ramsey configurations}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf}{OAI:A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/172.md.

Classification of finite Euclidean Ramsey configurations

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

A finite configuration is Euclidean Ramsey if every finite coloring of some sufficiently high-dimensional Euclidean space contains a monochromatic congruent copy at the original scale. The formalization gives the tensor-field classification for full-affine-span configurations, covers singleton and affine-span reductions, and proves that every Ramsey configuration is cospherical. It also proves that every nonempty subset of a finite transitive configuration and every nonempty set of at most five points on a circle is Ramsey. A further sufficient condition uses linear independence of the quadratic evaluation rows of a spherical configuration.

The formalization also gives spherical non-Ramsey examples: the specified twelve-point set on the unit circle has a fifty-color obstruction in every positive dimension, and a nine-point circle configuration built from algebraically independent parameters is non-Ramsey.

Comparator links

Result Comparator statement
Tensor-field classification of Ramsey configurations EuclideanRamsey.lean
Twelve-point spherical non-Ramsey example GrahamSpherical.lean
Ramsey property for at most five circle points EuclideanRamseyCircle.lean
Nine-point non-Ramsey circle configuration EuclideanRamseyNine.lean
Quadratic-independence criterion for spherical configurations EuclideanRamseyQuadratic.lean
Cosphericity of Ramsey configurations EuclideanRamseySpherical.lean
Ramsey property for subsets of finite transitive configurations EuclideanRamseyTransitive.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.