VibeMathedMath problems solved with AI
Sign in

Graham's spherical conjecture: every finite spherical set is Ramsey

A finite set PP in Euclidean space is Ramsey if for every number of colours kk there is a dimension nn such that every kk-colouring of Rn\mathbb R^n contains a monochromatic congruent copy of PP. Erdős, Graham, Montgomery, Rothschild, Spencer and Straus (1973) proved that every Ramsey set is spherical, that is, lies on some sphere. The converse became the central conjecture of the area, usually attributed to Graham: is every finite spherical set Ramsey? In particular, is every finite set of points on a circle Ramsey?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Euclidean Ramsey theory
Posed by
Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, Euclidean Ramsey theorems I, J. Combin. Theory Ser. A 14 (1973); later stated as a conjecture by Graham
Year posed
1973
Years open
53y
Solved
2026-09-20
Model
ChatGPT (version not stated)
Vendor
OpenAI
Collaborators
Dömötör Pálvölgyi
Verification
Unreviewed
Publication
Preprint
Significance
40 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims a seven-point cyclic set, for any transcendental radius parameter r>2r>2 (for example r=πr=\pi), that is not Ramsey: a finite colouring of Rn\mathbb R^n with no monochromatic congruent copy, in every dimension. Since the set lies on a circle, this disproves the conjecture that every spherical set is Ramsey. The colouring is non-constructive, built from a non-measurable derivation of R\mathbb R. The unverified appendix adds that almost every seven-point circular set is non-Ramsey and that the method cannot work below seven points. The OpenAI release's classification manuscript (23 September 2026) cites this paper as the disproof, adds a twelve-point spherical example by the same derivation-moment method, and proves every set of at most five circle points Ramsey; six circle points remain open between the two.

What the AI did

From the paper's AI disclosure: "This manuscript was written entirely by ChatGPT. I (Dömötör Pálvölgyi) contributed nothing to the proofs or ideas, I only gave suggestions about the presentation." The author's closing note says the construction was found by ChatGPT, that he had tried similar field-automorphism constructions himself without success, and that the appendix contains further results by ChatGPT he has not verified. The model version is not stated.

Verification

No independent mathematician has checked this yet. Read here on 7 October 2026 from the arXiv source: Theorem 1 states that for any transcendental r>2r>2 the seven points (0,0)(0,0) and (j,±j(2r−j))(j,\pm\sqrt{j(2r-j)}), j∈{1,3,4}j\in\{1,3,4\}, lie on the circle (x−r)2+y2=r2(x-r)^2+y^2=r^2 and are not Ramsey, which contradicts the conjecture as posed. The proof is one page: a field derivation DD with D(r)=1D(r)=1, weights 2,−2,2,−12,-2,2,-1 whose zeroth, first, second and mixed moments vanish, and a weighted energy identity ∑λp∥D(p′)∥2=−24/R\sum\lambda_p\|D(p')\|^2=-24/R invariant under congruence, closed by Rado's theorem. The moment conditions and the energy identity were recomputed by hand here and hold. The appeal to Rado and the existence of DD (which needs a transcendence basis) were read, not re-derived. The author states he contributed nothing to the proof; the appendix is marked unverified by him. No formal proof.

Sources

Changelog1 change

Discussion