The classification problem for finite Euclidean Ramsey sets

A finite set $A$ in Euclidean space is Ramsey if for every number of colors $r$ there is a dimension $D$ such that every $r$ -coloring of $\mathbb R^D$ , with no regularity assumed, contains a monochromatic congruent copy of $A$ at its original scale. Erdos, Graham, Montgomery, Rothschild, Spencer and Straus introduced the notion in 1973 and proved that every Ramsey set is spherical. Known Ramsey sets include nondegenerate simplices (Frankl-Rodl), sets with a soluble transitive isometry group and cyclic trapezoids (Kriz), and vertex sets of regular polytopes (Cantwell). Graham conjectured that every spherical set is Ramsey (1994); Leader, Russell and Walters conjectured that a set is Ramsey exactly when it embeds in a finite transitive set (2012). Which finite sets are Ramsey: is there a necessary and sufficient criterion?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Euclidean Ramsey theory
Posed by: Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, Euclidean Ramsey theorems I, J. Combin. Theory Ser. A 14 (1973)
Year posed: 1973
Years open: 53y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 45 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: a finite $A\subset\mathbb R^d$ of at least two points with full affine span is Ramsey iff some $(d+1)\times(d+1)$ matrix $P$ over $F\otimes_{\mathbb Q}F$ , with $F$ the field generated by the coordinates, satisfies $(p_i\otimes1)^TP(1\otimes p_i)=0$ for every $p_i=(1,a_i)$ and $m_F(P_{\alpha\beta})=\delta_{\alpha\beta}$ on the spatial block. Consequences: every subtransitive set and every set of at most five points on a circle is Ramsey; nine circle points with algebraically independent parameters, and twelve points forming three rotated squares, are not. The criterion is exact algebra over the coordinate field; the paper says it is not a procedure for deciding the property from numerical coordinates. The Leader-Russell-Walters disproof is a separate entry.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the introduction, Theorem 1.1 and the consequences section of the TeX source; the proof was not refereed. Lean-checked on the Comparator challenge EuclideanRamsey, listed in the release's formalization catalogue (OAI.EuclideanRamsey.classification, OAI/Combinatorics/EuclideanRamsey/Main.lean). Its statement: for an injective family of $s\ge2$ points in $\mathbb R^d$ , $d\ge1$ , with full affine span, being Ramsey (for every $r\ge2$ some $D\ge1$ such that every $r$ -coloring of $\mathbb R^D$ has a monochromatic family with the same pairwise distances) is equivalent to the existence of the tensor matrix over the coordinate field. That is Theorem 1.1. Further challenges for this family state cosphericity of Ramsey sets, the subtransitive and five-circle-point sufficiency results, the quadratic-independence criterion, the nine-point example (these five are not in the catalogue; their solution files exist at the pinned commit) and the twelve-point example (GrahamSpherical, catalogued). Statements read here; nothing rebuilt here.

The classification problem for finite Euclidean Ramsey sets

What was actually shown

What the AI did

Verification

Sources

Discussion