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

Claims a seven-point cyclic set, for any transcendental radius parameter $r>2$ (for example $r=\pi$ ), that is not Ramsey: a finite colouring of $\mathbb R^n$ 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 $\mathbb R$ . 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.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 20, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Graham's spherical conjecture: every finite spherical set is Ramsey · Graham's spherical Ramsey conjecture

Confidence: Not scored

Registry verification: unreviewed · preprint · candidate

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Attribution

VibeMathed
registry · event recorded by

ChatGPT (version not stated)
model · ai model contributor · OpenAI

Dömötör Pálvölgyi
human · human collaborator

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

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

specialist registry · passed

No independent mathematician has checked this yet. Read here on 7 October 2026 from the arXiv source: Theorem 1 states that for any transcendental $r>2$ the seven points $(0,0)$ and $(j,\pm\sqrt{j(2r-j)})$ , $j\in\{1,3,4\}$ , lie on the circle $(x-r)^2+y^2=r^2$ and are not Ramsey, which contradicts the conjecture as posed. The proof is one page: a field derivation $D$ with $D(r)=1$ , weights $2,-2,2,-1$ whose zeroth, first, second and mixed moments vanish, and a weighted energy identity $\sum\lambda_p\|D(p')\|^2=-24/R$ 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 $D$ (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.

Source ↗

Palvolgyi, A cyclic non-Ramsey heptagon (arXiv:2609.23327), the disproof of Graham's conjecture cited here

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, Euclidean Ramsey theorems I (1973)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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The Leader-Russell-Walters conjecture: is every Euclidean Ramsey set subtransitive?

witness · pending

paper link supplied by the VibeMathed registry.

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Lean: OAI/Combinatorics/SphericalRamsey/Main.lean (twelve-point non-Ramsey example)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean: OAI/Combinatorics/SphericalRamsey/Main.lean (twelve-point non-Ramsey example)

lean artifact · pending

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Compute record

No linked compute attempts recorded.

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VibeMathed record: Graham's spherical conjecture: every finite spherical set is Ramsey evidence for this event

This event attributed to Dömötör Pálvölgyi

Graham's spherical conjecture: every finite spherical set is Ramsey parent of this event

This event attributed to ChatGPT (version not stated)

The Leader-Russell-Walters conjecture: is every Euclidean Ramsey set subtransitive? evidence for this event

Palvolgyi, A cyclic non-Ramsey heptagon (arXiv:2609.23327), the disproof of Graham's conjecture cited here evidence for this event

Lean: OAI/Combinatorics/SphericalRamsey/Main.lean (twelve-point non-Ramsey example) evidence for this event

Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, Euclidean Ramsey theorems I (1973) evidence for this event

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