Generalised Prisms and Euclidean Ramsey Theory

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block_template_consequences: Deduces the geometric Ramsey and uniform-block conclusions for 1223333 and 12233333 from exact identified inputs.

conjecture_6: States the block sets conjecture as the paper poses it, with its definitions of template and block set, and the geometric Ramsey implication for a fixed template.

conjecture_8: States the paper's Conjecture 8, that adding one point off the hyperplane of a Ramsey set gives a Ramsey set, with the paper's remarks and the later proofs.

corollary_4: States the paper's pyramid corollary: a finite transitive base with one point added off its hyperplane is subtransitive, and subsoluble, hence Ramsey, when the base is subsoluble.

corollary_5: States the paper's corollary that two concentric regular polygons placed in parallel planes at any nonzero distance form a Ramsey set.

definitions: Proves the elementary group and geometric facts used in the generalized-prism construction.

examples: Proves the source examples of isosceles trapezia, lifted triangle midpoints and rationally rotated rectangles.

external_inputs: States the external soluble-group, spherical-necessity and uniform-block results without claiming their proofs.

isometric_extension: Proves that a finite isometric embedding extends to its ambient space after adding orthogonal coordinates.

lemma_2: States the paper's height-raising lemma: if Theorem 1 holds for the fixed sets at one positive height, it holds at every greater height.

lemma_3: States the paper's small-height lemma: Theorem 1 holds at the height equal to the distance between a point of X and a point of Y divided by the square root of n.

soluble_prime_degree: Proves the elementary prime-degree order bound used in place of a classification table in Theorem 7.

template_symmetry_qualification: Gives an exact counterexample to the claim that algebraically independent alphabet values force every template isometry to permute coordinates.

theorem_1: States the paper's main theorem: two finite sets on which one finite isometry group acts transitively form, at any nonzero height, a prism contained in a finite transitive set, whose group can be taken soluble when the first one is.

theorem_7: States the paper's template theorem: 1223333 has no transitive soluble symmetry group, but it embeds in 12233333, which has one.

two_color_observation: Completes the source sphere-growth argument for adjoining an apex above the circumradius, with explicit dimension and radius control.

zero_height_obstruction: Proves that an equilateral triangle together with its center is nonspherical, so the nonzero-height hypothesis is essential.

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Maria-Romina Ivan, Imre Leader and Mark Walters, Generalised Prisms and Euclidean Ramsey Theory, arXiv:2606.13472v1, submitted 11 June 2026 at 15:25:49 UTC (versioned record). The copy read for this card is the ten-page v1 PDF. The source snapshot records its identity, source reading, exact external inputs and provenance. No published-version comparison is asserted. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2606.13472), every other right reserved.

Main results. The paper's main theorem is Theorem 1 (p. 2): if one finite group $G$ of isometries of $\mathbb R^d$ acts transitively on each of two finite sets $X$ and $Y$, then for every $\lambda\ne0$ the generalised prism $(X\times\{0\})\cup(Y\times\{\lambda\})$ is contained, up to congruence, in a finite transitive set, and in a soluble one when $G$ is soluble, so that it is then Ramsey by Kříž's theorem. Its proof runs through Lemma 2 (p. 3), which raises the height, and Lemma 3 (p. 4), which gives arbitrarily small heights. Corollary 4 (p. 6) makes a pyramid over a finite transitive base subtransitive, and subsoluble, hence Ramsey, when the base is subsoluble; Corollary 5 (p. 6) makes two concentric regular polygons in parallel planes Ramsey. Section 3 (pp. 7–9) states the block sets conjecture of Leader, Russell and Walters as Conjecture 6 (p. 7) and proves Theorem 7 (p. 8): the template $1223333$ has no transitive soluble symmetry group but embeds in $12233333$, which has one. Section 4 (pp. 9–10) poses Conjecture 8 (p. 9), that one point added off the hyperplane of any Ramsey set gives a Ramsey set, and asks about two prism-type configurations that Theorem 1 does not cover.

Points read differently from the print. Each is recorded on the page named. Lemma 2's statement defines $Z'$ with $\lambda$ where $\lambda''$ is meant, and its soluble case needs a chosen soluble transitive group rather than the full symmetry group (Lemma 2). Lemma 3's proof asserts that $X$ and $Y$ have the same centre, which is neither true in general nor needed (Lemma 3). Corollary 4's printed proof covers the subsoluble clause only when a soluble group acts on the base itself; the corpus supplies the missing step (Corollary 4, isometric extension). The p. 7 remark that, for algebraically independent values, the geometric set of a template has only coordinate-permutation symmetries fails for $1122$ (counterexample); Theorem 7 uses only the coordinate action. Theorem 7's diagram repeats $\alpha^5$ where $\alpha^6$ is meant, and its group list names three groups as "neither". The p. 9 two-colour remark calls the second sphere blue where the argument forces it red (two-colour observation).

Corpus pages beyond the paper's labels. The pages definitions, isometric extension and prime-degree bound give elementary facts in the corpus's own proofs; examples, zero-height obstruction, two-colour observation and block consequences work out remarks the paper makes on pp. 2, 7–8 and 9; external inputs states the results of other papers used (Kříž's soluble-group theorem, the spherical-necessity theorem of Erdős et al., and the Leader–Russell–Walters block family). The source snapshot records the reading and these findings.

Read status. Claims checked: the statements of Theorem 1, Lemmas 2 and 3, Corollaries 4 and 5, Conjecture 6, Theorem 7 and Conjecture 8 were read clause by clause on the PDF, with their proofs read for structure. The external inputs are statements, not proofs, here.

Bibliography. Two of the paper's references are misprinted: source reference [5], Frankl–Rödl's A partition property of simplices in Euclidean space, is J. Amer. Math. Soc. 3 (1990), 1–7, DOI 10.1090/S0894-0347-1990-1020148-2, not the printed volume 136 and pages 119–127; source reference [6], Block sizes in the block sets conjecture, is Forum Math. Sigma 14 (2026), e67, DOI 10.1017/fms.2026.10212, not volume 16. The latter publisher record was checked; its proof is not imported into this unit.

Later work. Papers of August 2026 by Moore and by Mirabi prove Conjecture 8's conclusion for every finite Ramsey base; their standing is recorded on their own cards. See Conjecture 8. The paper was found on Maria-Romina Ivan's Cambridge publications page; no journal version was located in that search, which is not an exhaustive absence claim.

Bears on. #174 (characterise the Ramsey sets): Theorem 1 and Corollaries 4 and 5 give families of subsoluble, hence Ramsey, sets (generalised prisms on two orbits of one soluble finite isometry group, pyramids over finite transitive subsoluble bases, prisms on concentric regular polygons), a sufficient condition and not a characterisation; Conjecture 8 conjectures a closure property of the Ramsey sets, which later papers address; Conjecture 6 and Theorem 7 concern the block sets conjecture, which implies that every subtransitive set is Ramsey. The paper does not decide the problem.

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