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Claim. Dömötör Pálvölgyi, A cyclic non-Ramsey heptagon, arXiv:2609.23327v1 [math.CO], 20 September 2026, carded at palvolgyi_2026_cyclic_non_ramsey_heptagon. Its Theorem 1, paged at Theorem 1, states that for every transcendental $r>2$ the seven points

$$(0,0)\quad\text{and}\quad\bigl(j,\pm\sqrt{j(2r-j)}\bigr),\qquad j\in\{1,3,4\},$$

lie on the circle $(x-r)^2+y^2=r^2$ and form a set that is not Ramsey in the sense of Problem 174; $r=\pi$ is an instance. The proof follows the method by which Erdős, Graham, Montgomery, Rothschild, Spencer and Straus excluded nonspherical sets, with a derivation of $\mathbb R$ over $\mathbb Q$ in place of the squared norm: integer weights summing to zero are chosen so that the weighted sums of the points, of their outer products and of their derivation images all vanish, while the weighted sum of the squared derivation images is a fixed nonzero constant on every congruent copy in every dimension. Rado's inhomogeneous theorem then gives a finite coloring of $\mathbb R$ with no monochromatic solution of the resulting scalar equation, and composing it with the derivation energy colors every $\mathbb R^n$ without a monochromatic copy. The manuscript's disclosure says that ChatGPT wrote it and found the proof and its ideas, the author contributing only suggestions on the presentation and the closing remarks, so the human submitter is the claimant here and the system is named as the manuscript names it. An appendix headed as written by ChatGPT and unchecked by the author states further claims, among them that fixed weighted derivation identities cannot show non-Ramseyness for at most six concyclic points and that almost every seven-point subset of a circle is not Ramsey; the author's note also announces a proof that every finite transitive set is Ramsey, whose exposition it says is still being prepared. Those appendix statements and that announcement are not part of this claim.

Covers. Graham's conjecture is false: a finite spherical set need not be Ramsey. The result settles no other part of the characterization; in particular it says nothing about which spherical sets are Ramsey, and the author's note leaves the rival Leader–Russell–Walters conjecture open. The same conjecture is refuted, three days later, by the twelve-point set in OpenAI's accepted classification, whose criterion applies to every finite set.

Depends on. No page of this wiki.

Acceptance. None documented. The manuscript is an arXiv preprint with no journal record known here, the site's statement and remarks for the problem do not mention it, and its proof-claims thread carried no claim as of 6 October 2026; the author announced the result in the problem's discussion on the site on 23 September 2026, attributing the counterexample to ChatGPT 6 and linking the exposition. Nothing here has reviewed the proof. OpenAI's preprint of 23 September 2026 cites this theorem as the disproof of the spherical conjecture, which is a dated uptake and not a review. The claim is therefore claimed.