Simplices and Regular Polygonal Tori in Euclidean Ramsey Theory

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abelian_orbits: Expands the source’s simultaneous-diagonalization observation, treating fixed and zero coordinates and finite induced groups explicitly.

definitions: Fixes finite polygon products, squared-distance approximation, affine independence and the all-color Ramsey convention.

evidence/: Retains the independent review of the complete simplex-enclosure chain and Ramsey corollary at the declared external inputs.

external_inputs: Identifies the soluble-group and finite Gram interfaces and separates historical citations from unresolved source reconstructions.

lemma_10: Makes the source’s Schoenberg step explicit by using a uniform strict negative-type margin and the canonical finite Gram criterion.

lemma_3: Constructs a regular simplex in a product of regular polygons of any prescribed order, including order two.

lemma_4: Constructs a labeled affinely independent realization using regular-simplex factors with exactly one identified pair per factor.

lemma_7: Proves the polygon approximation with an explicit injectivity condition and records a counterexample to the printed unrestricted threshold.

proposition_11: Bounds the entire residual distance array, realizes it by an almost-regular simplex and adds it orthogonally to the approximation.

proposition_5: Combines the labeled distance realization and regular-simplex embeddings while keeping one common polygon order.

proposition_8: Chooses a common polygon radius and order for every coordinate projection and handles singleton projections explicitly.

ramsey_corollary: Deduces the all-color Ramsey property for simplex enclosures from Kříž’s exact transitive soluble-group theorem.

regular_expansions: Proves the product identity and affine independence of every positive regular expansion of a finite configuration.

theorem_2: Completes Karamanlis’s simplex enclosure theorem by the contraction, approximation and almost-regular correction chain.

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Miltiadis Karamanlis, Simplices and Regular Polygonal Tori in Euclidean Ramsey Theory, The Electronic Journal of Combinatorics 29(3) (2022), P3.66, DOI 10.37236/10944. The canonical attachment is the published eight-page article. Its first page records submission on 27 December 2021, acceptance on 3 July 2022 and publication on 23 September 2022, under CC BY 4.0. The publisher record and source record identify the exact version.

The article proves that every finite affinely independent configuration embeds, with all distances unchanged, into a finite product of vertex sets of regular polygons. The proof actually gives a common polygon order; the factor radii may differ. Independent cyclic rotations act transitively on this product, so it supplies an abelian, hence soluble, transitive enclosure. Kříž's theorem then gives an alternative proof that every simplex is Ramsey for every finite number of colors.

The complete main chain is retained in Theorem 2, with all essential same-paper steps:

• Lemma 3 embeds a regular simplex using two adjacent polygon vertices in each coordinate.
• Lemma 4 realizes every sufficiently near-regular distance array by a product of regular simplices. Its pair factors identify exactly one labeled pair each.
• Proposition 5 turns that realization into a polygon product of any prescribed common order.
• Lemma 7 rounds a finite subset of a line and places it on a large regular polygon. The strengthened parameter range is proved together with its strict error and injectivity bounds.
• Proposition 8 combines coordinate approximations with one common order and radius, including singleton projections.
• The regular-expansion identity proves the diagonal-product interpretation and affine independence.
• Lemma 10 contracts every simplex by a small positive regular squared-distance term. The full deduction uses the existing finite Gram criterion once, with an explicit uniform strict margin.
• Proposition 11 realizes the residual error array as an almost-regular simplex and adds its embedding orthogonally. The full deficit estimate justifies the source's tolerance $\delta=\alpha^2/n^2$.

The Ramsey consequence is a complete relative deduction from the canonical soluble-group theorem. The distinct introductory abelian-orbit observation is expanded through simultaneous unitary diagonalization. Altogether the unit contains eleven complete source-related or expanded deductions and one separate compilation counterexample to the printed approximation range. Exact outside interfaces and historical boundaries are stated separately; definitions and source metadata add no proof count.

The necessary substantive correction is explicit. Published Lemma 7 assumes only $n\ge2\pi n_0^3/\delta$, which need not leave enough vertices for an injection. For example, its allowed parameters can demand an injection of ten points into an eight-vertex polygon. We prove the sufficient integer choice $n\ge\max\{2,n_0,2\pi n_0^3/\delta\}$. This preserves every later existential conclusion. Other expansions separate diagonal cases from off-diagonal squared-distance identities, justify the common coordinate parameters and the strict residual inequality, and make Lemma 10's cited finite-dimensional deduction explicit. These are compilation explanations and repairs, not an author-issued erratum.

The earlier PDFs are preserved exactly as arXiv v1, 17 May 2021, v2, 13 June 2021 and v3, 8 July 2022. Each has seven pages. All 29 pages across the four versions were visually read. The arXiv record supplies their dates; publication does not make these different byte artifacts interchangeable. The published article (karamanlis_2022_simplices_regular_polygonal_tori.pdf) prints "© The author. Released under the CC BY license (International 4.0)." on its first page, the Creative Commons Attribution 4.0 license. The arXiv record (https://arxiv.org/abs/2105.07689, read 2026-10-02) names the Creative Commons Attribution 4.0 license for the arXiv v1 PDF (karamanlis_2022_simplices_regular_polygonal_tori_arxiv_v1.pdf). The arXiv record names the same license for the arXiv v2 PDF (karamanlis_2022_simplices_regular_polygonal_tori_arxiv_v2.pdf; arXiv:2105.07689). The arXiv record names the same license for the arXiv v3 PDF (karamanlis_2022_simplices_regular_polygonal_tori_arxiv_v3.pdf; arXiv:2105.07689).

The main arXiv label is Theorem 1.2. Its Lemmas 3.1, 3.2, 3.5 and 3.8 become published Lemmas 3, 4, 7 and 10; Propositions 3.3, 3.6 and 3.9 become Propositions 5, 8 and 11. Versions 1 and 2 use the earlier $n_0^3/n^2$ rounding display and incorrect strictly positive endpoint wording. Version 3 and the publication use $2n_0^2/n^2$, a nonnegative floor index and a possibly zero rounding error. Version 3 also adds the Matoušek–Rödl acknowledgment and expands the comparison with the original Frankl–Rödl proof. The common still-insufficient parameter threshold is distinguished from these author revisions. The full label, page, version and correction map is in the source record.

The printed bibliography truncates several ranges. The linked canonical records give Frankl–Rödl (1987), pp. 259–286, Frankl–Rödl (1990), pp. 1–7, and Kříž (1991), pp. 899–907. The cited Schoenberg article is pp. 522–536; its full original paper is not reviewed here. The source's quoted Frankl–Pach–Reiher–Rödl Lemma 4.9 proof is reconstructed as Lemma 4, without claiming coverage of the rest of that chapter.

This is a materially distinct simplex proof from the original exponential-density construction. The torus proof asserts no prescribed witness radius and does not give the later near-circumsphere conclusion. A subset's intrinsic circumradius can be smaller than the radius of its containing torus. The article's historical conjecture discussion does not establish a characterization of all spherical or Ramsey sets. Its precise positive class belongs with Problem 174, whose separate source record also traces Behague's later use of this construction. No problem status or later-paper proof is being inferred from that citation.