Transitive Sets in Euclidean Ramsey Theory

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binary_templates: The paper's unnumbered case m = 2 of Conjecture E: for the template of r ones and s twos, Ramsey's theorem gives a monochromatic block set of degree r + s with singleton blocks in every k-coloring of [2]^n, n large.

conjectures: States the paper's classification conjecture and its equivalent proposed sufficient conditions, preserving the fixed degree and uniformity quantifiers.

external_inputs: Lists the outside theorems the paper's proofs use, Ramsey's theorem, van der Waerden's theorem, a representation lemma and Sard's theorem, and the earlier Euclidean Ramsey results it recalls, with their corpus pages.

historical_questions: Preserves the 2010 manuscript's block, embedding and witness questions without treating its historical open labels as current certifications.

lemma_2_3: For algebraically independent reals alpha_1, ..., alpha_m, every copy of sX, s > 0, inside {alpha_1, ..., alpha_m}^(mn), X the permutations of (alpha_1, ..., alpha_m), is the image of an s^2-uniform block set; the proof needs m >= 3 and the statement fails for m = 2.

lemma_4_1: For a fixed cyclic k-gon in R^n, k >= 16, the cyclic k-gons that embed in R^n with all corresponding vertex distances equal and with plane not orthogonal to the fixed one form a set of measure zero.

minimal_witnesses: The paper's heuristic for X = {0,1}: the minimal k-Ramsey sets are those whose unit-distance graph is (k+1)-critical, odd cycles for k = 2, and the unique minimum-sized one is the regular simplex on k + 1 vertices.

orthogonal_representation_count: The step of the proof of Theorem 4.2 that a finite group has only countably many orthogonal representations up to orthogonal conjugacy, so the orthogonal groups have countably many finite subgroups up to conjugation.

proposition_2_1: The fixed-degree group conjecture C implies the scaled-power conjecture B: a monochromatic group line of fixed size d in G^n gives a monochromatic copy of X in the scaled power d^(-1/2) X^n.

proposition_2_2: The fixed-degree group conjecture C and the fixed-degree block permutation conjecture D are equivalent; the nontrivial direction applies C to the symmetric group and reads each permutation through its inverse.

proposition_2_4: The scaled-power conjecture B implies the uniform block-set conjecture F, through Lemma 2.3 applied to the permutation orbit of algebraically independent reals; with Propositions 2.1 and 2.2 it makes B to F equivalent.

source_corrections: Records the version whose labels and pages the card uses and the misprints and gaps the corpus reads in it: the missing m >= 3 in Lemma 2.3, the scale 1/sqrt(m) in Proposition 2.1, index slips, and an r for r^2 in Lemma 4.1.

template_substitution: The paper's unnumbered remark that Conjecture D on the alphabet [l] gives Conjecture E for a template of length l, by recoloring through the template's letters; the blocks, and so the degree and uniformity, are kept.

theorem_3_1: For the alphabet [3] and every template of r ones, s twos and one three, every k-coloring of [3]^n, n large, contains a monochromatic block set of a fixed degree with that template; the proof makes all blocks the same size, the first nontrivial cases of the paper's block-set conjectures.

theorem_4_2: For each k >= 16 the subtransitive cyclic k-gons form a set of measure zero, so some cyclic 16-gon is not subtransitive: not every spherical set embeds in a finite transitive set. The proof gives no explicit polygon.

three_value_orbits: The paper's consequence of the uniform blocks in the proof of Theorem 3.1: for distinct reals alpha, beta, gamma, the points of R^(r+s+1) with r coordinates alpha, s coordinates beta and one coordinate gamma form a Ramsey set.

transitive_sets_are_spherical: The paper's observation that the points of any finite transitive set lie on the surface of the unique smallest closed ball containing it, so every subtransitive set is spherical.

triangle_embedding: The paper's illustration that every triangle embeds in a finite transitive set: two parallel copies of a regular polygon, one rotated, chosen so that the two base vertices lie on the polygon.

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Imre Leader, Paul A. Russell and Mark Walters, Transitive Sets in Euclidean Ramsey Theory, Journal of Combinatorial Theory, Series A 119(2) (February 2012), 382–396, DOI 10.1016/j.jcta.2011.09.005.

The copy read for this card is the 20-page arXiv:1012.1350v1, submitted 6 December 2010. A separate 20-page author manuscript dated 22 November 2010 was also consulted. The arXiv PDF's displayed 2018 date is not a listed arXiv revision. The published 15-page text has not been acquired or compared; the result labels and page numbers in this unit refer to arXiv v1. See the exact source record and [[discrete_geometry/leader_2012_transitive_sets_euclidean_ramsey_theory/source_corrections|the corrections page]]. The arXiv record names arXiv's non-exclusive distribution license for the arXiv PDF (arXiv:1012.1350), every other right reserved. The author manuscript is the one 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 manuscript prints none on any of its 20 pages; the term is unstated.

The paper proposes (Conjecture A, p. 3) that a finite set is Ramsey if and only if it is subtransitive, that is, congruent to a subset of a finite transitive set. It reduces the "if" direction to equivalent combinatorial conjectures B--F, proves their first nontrivial cases, and proves that not every spherical set is subtransitive, so that Conjecture A differs from Graham's conjecture that the Ramsey sets are the spherical ones. It proves neither conjecture.

Results. Labels and pages are those of arXiv v1.

• Conjectures A--F (pp. 3--9), with the definitions of block sets and templates.
• Proposition 2.1 (p. 6): C implies B.
• Proposition 2.2 (p. 7, proof pp. 7--8): C and D are equivalent.
• Remark (p. 9): D and E are equivalent.
• Lemma 2.3 (p. 10, proof pp. 10--11): copies of $sX$ in the word cube over algebraically independent reals are $s^2$-uniform block sets; the proof needs $m\ge3$.
• Proposition 2.4 (p. 12): B implies F, closing the equivalence of B--F.
• Two-letter templates (p. 12): E and F hold for $m=2$.
• Theorem 3.1 (p. 12, proof pp. 12--14): E holds for $m=3$ and the templates $1^r2^s3$, with uniform blocks.
• Remark (p. 15): the permutation orbits of $r$ copies of $\alpha$, $s$ of $\beta$ and one $\gamma$ are Ramsey.
• Lemma 4.1 (p. 16): for $k\ge16$, cyclic $k$-gons at equal corresponding distances from a fixed one, in a non-orthogonal plane, form a null set.
• Theorem 4.2 (p. 17): for each $k\ge16$ the subtransitive cyclic $k$-gons have measure zero; some cyclic $16$-gon is not subtransitive. Its countability step is on its own page.
• Remarks on triangles in twisted prisms (p. 3), sphericity of transitive sets (pp. 3--4) and [[discrete_geometry/leader_2012_transitive_sets_euclidean_ramsey_theory/minimal_witnesses|minimal Ramsey sets for a unit pair]] (p. 18), and the [[discrete_geometry/leader_2012_transitive_sets_euclidean_ramsey_theory/historical_questions|Problems G, H, J, K and Conjecture I]] (pp. 15--19).

The external inputs and the corrections the corpus reads in the print have their own pages.

Read status. Claims checked: each result page's statement was read clause by clause against arXiv v1, and the proofs of Propositions 2.1, 2.2 and 2.4 and Lemmas 2.3 and 4.1 and Theorem 4.2 were read in full and followed; the proof of Theorem 3.1 was read for structure. Nothing here is independently reviewed.

Bears on. Problem 174: the paper's Conjecture A is a proposed answer to the problem's request to characterize the Ramsey sets, a rival to Graham's spherical conjecture. Theorem 4.2 shows that the two proposals differ, without proving any set non-Ramsey; the uniform blocks in the proof of Theorem 3.1 give new Ramsey sets (remark, p. 15). The paper proves neither proposal.

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