# Classification of finite Euclidean Ramsey configurations > An exact algebraic test tells which finite point sets must appear in one color in any coloring of space. Plain-English explainer, Lean status and sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-172/ Last updated: 2026-10-14 **An exact algebraic test tells which finite point sets must appear in one color in any coloring of space.** **Trust:** Formal proof listed · **Papers:** 1 · **First paper:** September 23, 2026 · **Field:** Combinatorics · **Entry:** 172 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. A finite set of points is Ramsey if every coloring of a high-dimensional space with finitely many colors contains a one-color copy of it. This paper proves an exact algebraic test for which sets are Ramsey. The test shows that every set of at most five points on a circle is Ramsey. That disproves a 2012 conjecture of Leader, Russell and Walters. ## At a glance - **First step:** 1973 (Erdős, Graham, Montgomery, Rothschild, Spencer and Straus) - **Significance:** Major. It answers the central classification question of Euclidean Ramsey theory, open since 1973. - **Trust:** Formal proof listed - **Field:** Combinatorics (1 paper) ## The explanation Color every point of space with one of three colors. Must some pair of points exactly one unit apart share a color? In ordinary three-dimensional space, yes. Take the four corners of a regular tetrahedron with all six edges one unit long. Four corners and three colors means two corners match, and they are one unit apart. With more colors, you just move to more dimensions, where larger such shapes fit. A finite set of points is called Ramsey when this always works. For any number of colors, some high enough dimension forces a copy of the set in a single color[1]. The copy must be congruent: every distance stays exactly the same, at the original scale. The coloring can be completely arbitrary, with no smoothness of any kind[2]. So the two-point set above is Ramsey. Not every set is. Three points evenly spaced on a line are not Ramsey. In 1973, Erdős, Graham, Montgomery, Rothschild, Spencer and Straus started this subject. They proved that every Ramsey set must lie on a sphere[3]. Three points on a line lie on no sphere. Graham later conjectured that lying on a sphere is enough[4]. Frankl and Rödl proved that every simplex is Ramsey, a simplex being a triangle, tetrahedron or higher version[5]. A second guess came from Leader, Russell and Walters in 2012. A set is transitive when its symmetries can move any of its points to any other, like a regular polygon's corners. They conjectured that a set is Ramsey exactly when it fits inside some finite transitive set[6]. Then, recently, Pálvölgyi found seven points on a circle that are not Ramsey[7]. That disproved Graham's conjecture. This paper proves an exact test for every finite set[8]. The test is an algebraic condition on the coordinates of the points. Lying on a sphere means the points satisfy one equation. The paper’s condition asks for a stronger form of that equation, one that keeps two copies of the coordinates apart. That stronger form is exactly what separates spheres from Ramsey sets[9]. The test works with exact algebra. It is not a recipe for deciding from rounded decimal coordinates[10]. The test settles concrete cases. Every set that fits inside a finite transitive set passes[11]. So does every set of at most five points on a circle[12]. That includes every four-sided shape whose corners lie on a circle. In 2011, Leader, Russell and Walters had found such kite shapes that fit in no finite transitive set. Those kites are Ramsey, so their 2012 conjecture fails[13]. The single difficulty was getting an exact copy from an arbitrary coloring. One direction is easier. When the condition fails, algebra builds a coloring that avoids the set in every dimension[14]. The other direction is hard. A wild coloring has no continuity, so a copy that is almost right is worth nothing[15]. The proof turns the algebra into two lists of numbers with matching distances. Then it uses tools from infinite combinatorics to land a copy at exactly the right scale. How sure is this? A computer proof checker called Lean has checked formal versions of the main results. They include the exact test, the fact that every Ramsey set lies on a sphere, and the transitive and five-point cases. They also include two sets that fail: a twelve-point set and a nine-point set on a circle. No specialist review of the paper has been published. **What it doesn't show.** This does not give a way to run the test from rounded decimal coordinates. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Major if the proof holds. Lean has checked formal versions of the classification and its main consequences. No specialist review is published. ## Words used - **Ramsey set:** A finite point set that appears in one color, at its original size, in every finite coloring of some high-dimensional space. - **Congruent copy:** A copy of a shape with every distance exactly the same. - **Spherical:** Lying on the surface of some sphere. - **Transitive set:** A finite point set whose symmetries can move any point to any other. - **Simplex:** A triangle, a tetrahedron, or the same shape in higher dimensions. - **Lean:** A computer program that checks each step of a formally written proof. ## The original paper - [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf) - September 23, 2026 · Formal proof listed - [Lean notes](https://github.com/openai/math/blob/main/lean/docs/172.md) ## Quoted lines from the paper (15) [1] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > A finite nonempty set $A$ in a Euclidean space is \emph{Ramsey} if, for every integer $r\ge2$, there is an integer $D\ge1$ such that every map $c:\R^D\to\{1,\ldots,r\}$ is constant on a congruent copy of $A$. [2] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > The coloring is an arbitrary function. [3] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus introduced Euclidean Ramsey theory and proved that every finite Ramsey set is spherical. [4] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus introduced Euclidean Ramsey theory and proved that every finite Ramsey set is spherical. Graham later conjectured that sphericity is also sufficient. [5] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > Major positive results established the Ramsey property for nondegenerate triangles and then all nondegenerate simplices, by Frankl and R\"odl; finite sets with a soluble transitive isometry group and cyclic trapezoids, by K\v{r}\'i\v{z}; and vertex sets of regular polytopes, by Cantwell. [6] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > Leader, Russell and Walters proposed a different characterization: a finite set is Ramsey exactly when it embeds in a finite transitive Euclidean set. [7] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > P\'alv\"olgyi has recently given a seven-point circle configuration that is not Ramsey, disproving the spherical conjecture. [8] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > Our main theorem gives a necessary and sufficient algebraic condition for an arbitrary finite Euclidean configuration. [9] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > This is the distinction between sphericity and the Ramsey property that the classification detects. [10] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > It is not a procedure for recovering those relations from numerical approximations to arbitrary real coordinates. [11] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > Every subtransitive set satisfies the tensor condition. Another sufficient condition is that evaluation at the points of a spherical set gives independent functionals on the space of quadratic polynomials. [12] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > It applies to every nonempty set of at most five points on a circle, and hence to every cyclic quadrilateral. In Corollary …, we combine this result with the non-subtransitive cyclic kites of Leader, Russell and Walters. [13] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > In Corollary …, we combine this result with the non-subtransitive cyclic kites of Leader, Russell and Walters. Those kites are Ramsey, disproving the necessity direction of their proposed characterization and their non-Ramseyness conjecture for the same family. [14] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > If no such tensor exists, algebraic separation produces a finite coloring that avoids $A$ in every dimension. [15] From [A classification of finite Euclidean Ramsey configurations](https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf): > For sufficiency, the difficulty is to convert the tensor cancellations into a congruent copy at scale one: arbitrary colorings provide no continuity with which to absorb approximation errors. The proof has three stages. ## Other sources - [P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus, “Euclidean Ramsey theorems. I”, Journal of Combinatorial Theory, Series A 14(3) (1973), 341–363](https://doi.org/10.1016/0097-3165(73)90011-3) - [R. L. Graham, “Recent trends in Euclidean Ramsey theory”, Discrete Mathematics 136 (1994), 119–127](https://doi.org/10.1016/0012-365X(94)00110-5) - [P. Frankl and V. Rödl, “A partition property of simplices in Euclidean space”, Journal of the AMS 3(1) (1990), 1–7](https://doi.org/10.1090/S0894-0347-1990-1020148-2) - [I. Leader, P. A. Russell and M. Walters, “Transitive sets in Euclidean Ramsey theory”, Journal of Combinatorial Theory, Series A 119(2) (2012), 382–396](https://arxiv.org/abs/1012.1350) - [I. Leader, P. A. Russell and M. Walters, “Transitive sets and cyclic quadrilaterals”, Journal of Combinatorics 2(3) (2011), 457–462](https://arxiv.org/abs/1012.5468) - [D. Pálvölgyi, “A cyclic non-Ramsey heptagon”, arXiv:2609.23327 (2026)](https://arxiv.org/abs/2609.23327v1) - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [The hypercube Ramsey conjecture](https://closedform.nishachay.tech/e/openai-math-2026-171/) - Combinatorics - [Sharp logarithmic exponents for off-diagonal Ramsey numbers](https://closedform.nishachay.tech/e/openai-math-2026-170/) - Combinatorics - [Superexponential van der Waerden numbers](https://closedform.nishachay.tech/e/openai-math-2026-160/) - Combinatorics - [Cycle–clique Ramsey numbers](https://closedform.nishachay.tech/e/openai-math-2026-189/) - Combinatorics - [Counterexamples to Ryser’s covering conjecture](https://closedform.nishachay.tech/e/openai-math-2026-162/) - Combinatorics - [Combinatorics](https://closedform.nishachay.tech/science/mathematics/combinatorics/) - Field · 37 results - [Ramsey theory](https://closedform.nishachay.tech/topic/ramsey-theory/) - Topic · 5 results - [Graph theory](https://closedform.nishachay.tech/topic/graph-theory/) - Topic · 40 results - [Open since the 1970s](https://closedform.nishachay.tech/open-since/1970s/) - Era · 63 results - [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 242 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20172%5D%20)