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

The manuscript proves that every nonempty set of at most five points on a circle is Ramsey, so every cyclic quadrilateral is Ramsey; in particular the kites $K_a$ with $a\in(-1,1)$ transcendental are Ramsey. With Leader, Russell and Walters' theorem that these kites are not subtransitive, this disproves the necessity direction of Conjecture A and disproves Conjecture 3. The sufficiency direction (subtransitive implies Ramsey) is proved in the same paper, so the conjecture fails in one direction only. The argument depends on the classification theorem of the same manuscript and on the cited non-subtransitivity result, which is not reproved.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. 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: The Leader-Russell-Walters conjecture: is every Euclidean Ramsey set subtransitive? · Leader-Russell-Walters conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the consequences section (the five-circle-point corollary and the kite corollary) of the TeX source; the proof was not refereed. Lean: the challenge EuclideanRamseyCircle (OAI.EuclideanRamsey.CircleConsequence.at_most_five_circle_points_ramsey, solution module OAI/Combinatorics/EuclideanRamsey/Circle.lean) is not in the release's formalization catalogue; its JSON and solution file exist at the pinned commit. Statement read here: every injective family of one to five points of $\mathbb R^2$ equidistant from some center is Ramsey. That is the Ramsey half of the disproof only. The non-subtransitivity of the kites is imported from Leader, Russell and Walters (2011, Corollary 2) and is not formalized, so no Lean statement asserts the disproof itself. Not rebuilt here. Listed as Unreviewed rather than Lean-checked because its formal statement covers only the Ramsey half of the disproof.