Nearly All Known Euclidean Ramsey Sets Are Subsoluble

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definitions: Defines the two orbit notions, proves affine-group solubility, and records why subsoluble configurations are Ramsey under Kříž's theorem.

external_inputs: Identifies the Kříž, Karamanlis and regular-polytope inputs and separates them from the paper's complete local constructions and historical context.

lemma_2_1: Proves that Cartesian products of subsoluble configurations are subsoluble, with the Euclidean product action made explicit.

lemma_3_1: Embeds every two-distinguished-coordinate signed permutation set in a soluble orbit under a wreath product with an affine group.

lemma_4_2: Constructs a transitive soluble wreath-product enclosure from a soluble two-orbit subgroup with orbit-stabilizer ratio at most two.

open_questions: Records the paper's five 2025 questions while separating subsoluble enclosures, Ramsey consequences, and present-day status evidence.

regular_polytope_actions: Gives explicit transitive soluble isometry actions for polygons, simplices, cubes, orthoplexes and the icosahedron.

source_corrections: Distinguishes all three arXiv versions, the deleted simplex proof, compilation-supplied repairs, and dated acceptance evidence.

theorem_1_5: Collects the coordinate-permutation, regular-polytope and isosceles trapezium results at their exact proved scopes.

theorem_1_6: Deduces subsolubility for the four permutation families through a signed orbit enclosure and exact fixed-coordinate embeddings.

theorem_4_1: Combines explicit soluble actions, the finite regular-polytope classification, and the two-orbit dodecahedron enclosure.

theorem_5_1: Deforms a symmetric cyclic trapezium to two regular polygon orbits and restores all six distances by a two-layer product construction.

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Natalie Behague, Nearly all known Euclidean Ramsey sets are subsoluble, arXiv:2510.15677v3, uploaded 4 December 2025 and dated 5 December 2025 on its title page, arXiv record.

The selected 12-page v3, 178,285 bytes, is the latest arXiv version and contains substantive corrections to two group actions. The materially distinct 15-page v1 and 12-page v2 are retained under version-specific names. Version 1 contains a later-deleted simplex proof with unresolved steps and receives no complete-proof credit. The exact comparison is in [[discrete_geometry/behague_2025_nearly_all_known_euclidean_ramsey_sets_subsoluble/source_corrections|the version and correction ledger]] and the source record. The arXiv record names the Creative Commons Attribution-ShareAlike 4.0 license for the v3 file behague_2025_nearly_all_known_euclidean_ramsey_sets_subsoluble.pdf (arXiv:2510.15677). The same record names the Creative Commons Attribution-ShareAlike 4.0 license for the v1 file behague_2025_nearly_all_known_euclidean_ramsey_sets_subsoluble_arxiv_v1.pdf. The same record names the Creative Commons Attribution-ShareAlike 4.0 license for the v2 file behague_2025_nearly_all_known_euclidean_ramsey_sets_subsoluble_arxiv_v2.pdf.

On 5 September 2026 the official Combinatorial Theory Accepted Papers page indexed Behague and this exact title. No deposited journal article, volume, issue, article number, journal PDF or publisher pagination was located, and the arXiv entry has no journal reference. The author's official papers page, with modification metadata from 13 August 2026, still labeled the paper “Submitted.” The evidence therefore supports accepted for publication, not publication; the canonical source remains arXiv v3.

A finite Euclidean configuration is soluble when a soluble isometry group acts transitively on it, and subsoluble when it embeds isometrically in a finite soluble configuration. Kříž's theorem makes every subsoluble configuration Ramsey. The exact definitions, affine-group calculation and Ramsey implication are in the preliminaries.

The complete v3 local chain is reconstructed as follows:

• Lemma 2.1 proves Cartesian-product closure and the rectangular-parallelepiped corollary.
• Lemma 3.1 constructs a transitive soluble signed-coordinate orbit under $C_2\wr\operatorname{AGL}(1,p)$, including repeated and zero parameters.
• Theorem 1.6 embeds four exact coordinate-multiplicity families into that orbit.
• Lemma 4.2 turns a suitable soluble two-orbit subgroup into a transitive soluble enclosure by a second wreath product.
• The explicit regular-polytope actions cover the infinite families and the icosahedron. Theorem 4.1 adds the two-orbit dodecahedron construction and classifies the result relative to the external finite convex regular-polytope list.
• Theorem 5.1 proves the symmetric isosceles-trapezium result through a continuous cyclic deformation and an exact two-layer distance restoration.
• Theorem 1.5 collects the three principal classes at the source's exact scopes.

The external-input page identifies Kříž's soluble-orbit theorem, Karamanlis's simplex enclosure, and the finite regular-polytope classification with the symmetry-group structures that Behague takes from Wilson's Table 2.1. It separates those results from the same-paper arguments and from historical or open-question context.

Questions 6.1–6.5 records the five subsolubility questions at the manuscript's dated 2025 scope, including the two-orbit enclosure question and the arbitrary coordinate-multiplicity question. It does not certify their later status.

The paper's title and abstract use “nearly all known” as a dated 2025 comparison. The reconstruction does not claim a current exhaustive inventory, novelty or priority; prove subsolubility for the 120-cell or 600-cell; settle the spherical or transitive-set conjectures; or answer Problem 174.

Bears on. #174.