Transitive sets and cyclic quadrilaterals

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conjecture_3: Records the paper's unproved Ramsey conjecture for its explicit cyclic kite.

corollary_2: Gives a symmetric four-point cyclic set that cannot embed in any finite transitive set.

generic_consequence: Expands the source's measure-zero consequence of the transcendental-parameter theorem.

lemma_4: Encodes quadrilateral parameters by an algebraic determinant polynomial and proves nonvanishing in the second parameter.

parameter_projection: Records the affine parameter convention and the geometric reductions to irreducible orthogonal representations.

theorem_1: Proves that a cyclic quadrilateral with the stated transcendental affine parameter cannot embed in any finite transitive set.

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Imre Leader, Paul A. Russell and Mark Walters, Transitive sets and cyclic quadrilaterals, Journal of Combinatorics 2 (2011), no. 3, 457--462, DOI 10.4310/JOC.2011.v2.n3.a6.

The paper separates spherical configurations from configurations that embed in finite transitive sets. Its main theorem uses the two affine parameters of a cyclic quadrilateral. Outside a countable union of algebraic exceptional sets, those parameters cannot occur in an orbit of a finite orthogonal group. In particular, the paper gives an explicit one-parameter family of cyclic kites that do not embed in any finite transitive set.

Source versions

The copy read for this card is the published version, six physical pages, printed pp. 457--462, read in full. Result-page citations use its printed pagination.

The five-page author PDF, linked from Mark Walters's publications page, is dated 25 December 2010. The arXiv record identifies the five-page arXiv v1 PDF as the sole version, submitted 25 December 2010. Its title-page date of 16 September 2018 comes from a later rendering of undated v1 TeX; it is not a second mathematical revision.

All three PDFs were visually read in full. The two five-page versions have the same mathematical text. The published version keeps the statements and proofs, updates the companion-paper reference from submitted to its 2012 publication, and makes small copy edits. In particular, it corrects the preprint's Conjecture 3 variable from $\alpha$ to $a$. All three versions retain a reversed existential sentence in the proof of Lemma 4: after asking for a value at which $P$ is nonzero, the text says that a nonzero kernel vector exists. The next sentence and the three cases use the required negation. The Lemma 4 page records the corrected logic. No notice is printed in the published PDF beyond the journal header and the footer "arXiv: 1012.5468", the Crossref record for DOI 10.4310/JOC.2011.v2.n3.a6 (read 2026-10-02) names no license, and the publisher's pages at intlpress.com could not be read on 2026-10-02 (HTTP 403); the term is unstated. The arXiv record names arXiv's non-exclusive distribution license for the arXiv v1 PDF (arXiv:1012.5468), every other right reserved. The author PDF is the authors' own copy of the arXiv submission, linked from an author's publications page (https://webspace.maths.qmul.ac.uk/m.walters/papers.html, read 2026-10-02), which states no copyright, license or terms, and the file prints none; the term is unstated.

Compiled results

• Parameters, projections and transitive spheres records the affine parameter convention, proves its compatibility with orthogonal decompositions, and proves the cyclicity observation for finite transitive sets.
• Lemma 4 (p. 459) gives the determinant polynomial for a nontrivial irreducible representation of a finite group and shows that it is not identically zero in the second parameter when $\alpha\ne0,1$.
• Theorem 1 (p. 458) proves that four distinct points on a circle with parameters $\alpha\ne1$ and $\beta$ transcendental over $\mathbb Q(\alpha)$ do not embed into a transitive set; the page also records the paper's remark that the condition $\alpha\ne1$ is necessary.
• Corollary 2 (p. 458) gives the explicit cyclic kites, one for each transcendental $a$, and computes their parameters.
• The almost-every consequence (p. 458, unnumbered) states the paper's almost-every claim for a stated measure and proves it, the paper calling the deduction routine.
• Conjecture 3 (p. 458) records the separate, unproved assertion that the explicit kites are not Ramsey.

The paper describes its examples as the first explicit spherical sets known at publication not to embed in a transitive set. That is a historical source claim, not a current priority finding made by this compilation. Failure to embed in a transitive set does not itself prove failure to be Ramsey; that separate step is Conjecture 3.

Bears on.

• Problem 174: the problem asks for a characterization of the Ramsey sets. Theorem 1 and Corollary 2 give spherical four-point sets that do not embed in any finite transitive set, so the class proposed by Graham (spherical sets) and the class proposed by Leader, Russell and Walters (subsets of finite transitive sets) differ. The paper proves nothing about whether these sets are Ramsey; Conjecture 3 states that the kites of Corollary 2 are not.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.