Palvolgyi 2026 nearcircumsphere ramsey theorem solvable transitive configurations

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Dömötör Pálvölgyi, A nearcircumsphere-Ramsey Theorem for Solvable Transitive Configurations, arXiv:2608.10865v1 [math.CO], 11 August 2026, 27 pp. (ELTE Eötvös Loránd University and Alfréd Rényi Institute of Mathematics, Budapest.) A revised version, v2 of 3 September 2026, exists and is the one the problem page cites; it is not held.

The retained folder-name PDF is the arXiv-generated PDF of v1, 27 pages with a text layer and the watermark "arXiv:2608.10865v1 [math.CO] 11 Aug 2026". All labels and page numbers below are v1's; no v2 label, page number or appendix is attached to this file. Provenance: retained from the repository's survey download set of September 2026; the file is arXiv's PDF of the record https://arxiv.org/abs/2608.10865v1, and the download date was not recorded; 579,288 bytes. The arXiv record (https://arxiv.org/abs/2608.10865, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Read status. Claims checked: Definition 1, Theorem 2, Corollary 3 and Theorem 4 were read clause by clause in the text layer (pp. 2--4), together with the introduction, the outline (p. 4), the statements of Theorem 11 and Proposition 13 (pp. 11--12), the short proof of Theorem 2 from them (p. 24) and the concluding remarks (pp. 24--25); the proofs in Sections 2--6 were not read.

Contents

The problem page states Theorem 2 as the author's claim, pending reconstruction; the statements below are v1's.

• Definition 1 (p. 2): for a finite spherical set $P$ with circumradius $\rho$ (the radius about the unique equidistant center in $\operatorname{aff}P$), $P$ is sphere-Ramsey if for every $r$ some sphere $\mathbb S^n_R$ forces a monochromatic copy of $P$ under every $r$-coloring; circumsphere-Ramsey if some $\mathbb S^n_\rho$ does; and nearcircumsphere-Ramsey (ncs-Ramsey) if for every $r\in\mathbb N$ and $\varepsilon>0$ some sphere $\mathbb S^n_{\rho+\varepsilon}$ forces a monochromatic copy of $P$ under every $r$-coloring. Then circumsphere-Ramsey $\Rightarrow$ ncs-Ramsey $\Rightarrow$ sphere-Ramsey $\Rightarrow$ Ramsey (p. 2). The paper attributes to Graham, through Reiher [40], the conjecture that all spherical sets are ncs-Ramsey, and records that two-point sets (Graham, Lovász) and simplices (Matoušek and Rödl) are ncs-Ramsey, while Kříž's proof gives sphere-Ramsey for solvable transitive sets (p. 2).
• Theorem 2 (p. 3): if $P$ is solvable transitive then $P$ is ncs-Ramsey; explicitly (p. 3), "if $P$ is a finite spherical set with circumradius $\rho$ and admits a solvable group of isometries that acts transitively on $P$, then for every $r\in\mathbb N$ and every $\varepsilon>0$ there is a dimension $n$ such that every $r$-coloring of $\mathbb S^n_{\rho+\varepsilon}$ contains a monochromatic congruent copy of $P$." The paper notes (p. 3) that this does not automatically extend to solvable subtransitive sets, and that the radius is sharp: apart from the one-point case, no transitive spherical configuration is circumsphere-Ramsey, as coloring each point of a sphere according to the sign of its first nonzero coordinate shows.
• Corollary 3 (p. 3): every regular polygon is ncs-Ramsey.
• Theorem 4 (Kneser-shift theorem, p. 4): for every prime $p$, $\chi(\mathrm{KSh}_p(n,k))\to\infty$ as $n-pk\to\infty$, where the vertices of $\mathrm{KSh}_p(n,k)$ are the ordered $(p-1)$-tuples of pairwise disjoint $k$-subsets of $[n]$ and $(A_1,\dots,A_{p-1})$ is adjacent to $(A_2,\dots,A_p)$ when $A_1,\dots,A_p$ are pairwise disjoint; proved through a special case of Ziegler's $\mathbb Z_p$-Tucker lemma. Whether the primality of $p$ is needed is left open.
• Structure (outline p. 4; proof of Theorem 2 on p. 24): Section 2 presents the ideas for the equilateral triangle; Section 3 develops dense block maps and the geometric reduction; Section 4 proves Theorem 4; Section 5 turns it into dense simultaneous rotation systems; Section 6 handles cyclic extensions and solvable groups. Theorem 11 (Dense block theorem, p. 11) states that a finite solvable group acting on a finite set has the dense block property $\mathrm{DB}(G\curvearrowright X)$, and Proposition 13 (Geometric reduction, p. 12) states that a spherical set $P$ with a finite transitive isometry group $G$ satisfying $\mathrm{DB}(G\curvearrowright P)$ is ncs-Ramsey; the proof of Theorem 2 applies Theorem 11 to the permutation image of the isometry group and then Proposition 13 (which it calls Theorem 13).
• Concluding remarks (pp. 24--25): seven open questions, among them whether Theorem 2 holds without solvability, whether every solvable subtransitive set is ncs-Ramsey, whether Theorem 4 holds for composite $p$, and density and canonical versions. The paper closes (pp. 25--26) with a statement that ChatGPT was used heavily in its preparation: none of the main ideas behind the ncs-Ramsey theorem came from it, but it contributed significantly to the proof of Theorem 4 and to improving the bounds, and it was used to rule out incorrect proof approaches, to check arguments, to draft and edit the text and to locate references.

Compiled scope

The introduction (pp. 1--4), the statements of Theorem 11 and Proposition 13, the proof of Theorem 2 from them (p. 24) and the concluding remarks (pp. 24--25) were read; Sections 2--6 (pp. 5--24), which carry the proofs of Theorems 4 and 11 and Proposition 13, were not read. Nothing here is independently reviewed, and v2 was not compared.

Bears on. #174, as the held copy (v1) of the preprint whose Definition 1 and Theorem 2 the page cites from v2 as a claimed near-circumsphere strengthening of Kříž's theorem for solvable transitive configurations, pending reconstruction there.