pith. sign in

arxiv: 2606.13472 · v1 · pith:F4DRWRWGnew · submitted 2026-06-11 · 🧮 math.CO · math.GR

Generalised Prisms and Euclidean Ramsey Theory

Maria-Romina Ivan , Imre Leader , Mark Walters This is my paper

Pith reviewed 2026-06-27 06:19 UTC · model grok-4.3

classification 🧮 math.CO math.GR
keywords euclidean ramsey theorygeneralised prismsisometry groupssoluble groupstransitive actionsramsey setsfinite symmetry
0
0 comments X

The pith

If two sets each admit a finite transitive isometry group then their prism in one higher dimension sits inside a finite set with such a group, preserving solubility.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to combine two sets X and Y, each equipped with its own finite isometry group acting transitively, into a prism in one higher dimension that is contained in a larger finite set admitting its own transitive isometry group. Solubility of the groups is inherited by the new group. Because Kriz proved that any set with a soluble transitive symmetry group is Ramsey, the construction produces new examples of Ramsey sets. Classifying which finite point sets are Ramsey under arbitrary finite colorings of Euclidean space remains a central open question. The prism method therefore supplies a concrete generation rule for enlarging the known collection of such sets.

Core claim

If a finite group of isometries of R^d acts transitively on a set X and also on a set Y, then the prism formed by X together with a translate of Y perpendicular to R^d is contained in a finite set in R^{d+1} on which a group of isometries acts transitively; moreover the new group remains soluble whenever the original groups are soluble.

What carries the argument

The generalised prism operation, which lifts two transitive orbits into a single transitive orbit on a finite superset in one higher dimension while preserving finiteness and solubility of the acting isometry group.

If this is right

  • Iterating the prism operation on known base cases produces Ramsey sets in arbitrarily high dimensions.
  • The class of sets possessing a soluble transitive isometry group is closed under the prism construction.
  • Any set obtained by finitely many prism combinations of soluble-symmetric sets is itself Ramsey.
  • The construction supplies an explicit algebraic method for proving that additional geometric configurations are Ramsey without constructing colorings directly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The operation could be used to generate infinite families of Ramsey sets whose minimal embedding dimensions increase without bound.
  • It remains open whether every Ramsey set arises from such iterated prism constructions starting from low-dimensional examples.
  • The same lifting technique might apply to other symmetry notions, such as affine or projective transformations, though the paper does not address this.

Load-bearing premise

The finite isometry groups of the original sets X and Y can always be extended or combined so that they act transitively on some finite superset containing the prism.

What would settle it

An explicit pair of sets X and Y, each admitting a finite soluble transitive isometry group, such that no finite superset of their prism admits any transitive isometry group at all.

read the original abstract

A finite subset $X$ of $\mathbb R^d$ is called Ramsey if for every $k$ there exists an $n$ such that whenever $\mathbb R^n$ is $k$-coloured there exists a monochromatic congruent copy of $X$. K\v r\'i\v z showed that if there is a soluble group of symmetries of $X$ that acts transitively on $X$, then $X$ is Ramsey. Determining which sets are Ramsey is a major unsolved problem. In this paper we show that if there is a finite group of isometries of $\mathbb R^d$ that acts transitively on a set $X$, and also on a set $Y$, then the `prism' formed by $X$ and $Y$ in $\mathbb R^{d+1}$ (meaning the set $X$ together with a translate of $Y$ in the direction perpendicular to $\mathbb R^d$) is itself contained in a finite set on which a group of isometries acts transitively. Moreover, if the initial group of isometries is soluble then so is the final group. This provides a new tool for generating Ramsey sets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript proves that if a finite group of isometries of R^d acts transitively on sets X and Y, then the prism X ∪ (Y + e_{d+1}) in R^{d+1} is contained in a finite superset Z that admits a transitive action by a finite isometry group H (soluble whenever the original group is). The result is positioned as a tool for constructing new Ramsey sets via Kříž's theorem on soluble transitive symmetry groups.

Significance. If correct, the construction supplies a systematic method for producing larger finite sets with the transitivity and solubility properties required by Kříž's theorem, thereby enlarging the known class of Euclidean Ramsey sets without introducing new parameters or unbounded constructions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending acceptance. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct constructive theorem: given finite isometry groups acting transitively on X and Y in R^d, an explicit finite superset Z in R^{d+1} is built containing the prism X ∪ (Y + e_{d+1}) together with a new finite (soluble) isometry group H acting transitively on Z. No equations, fitted parameters, or self-referential definitions appear. The result is an existence claim whose proof supplies the group H explicitly; Kříž's theorem is invoked only as an external application to obtain Ramsey sets, not as load-bearing justification for the construction. No self-citation chains or ansatzes reduce the central claim to its inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure existence proof in finite group actions on Euclidean space. No numerical parameters are fitted, no new entities are postulated, and the background axioms are the standard axioms of Euclidean geometry and finite group theory.

axioms (1)
  • standard math Standard axioms of Euclidean isometries and finite group actions
    Invoked implicitly when defining transitive actions and prisms in R^{d+1}.

pith-pipeline@v0.9.1-grok · 5732 in / 1311 out tokens · 15176 ms · 2026-06-27T06:19:02.171244+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references

  1. [1]

    P. Erd. Euclidean. Journal of Combinatorial Theory, Series A , volume=

  2. [2]

    Frankl and V

    P. Frankl and V. R\"odl , year = 1986, journal =. All triangles are

    1986

  3. [3]

    Frankl and V

    P. Frankl and V. R\"odl , year = 1990, journal =. A partition property of simplices in

    1990

  4. [4]

    Karamanlis , year = 2022, journal =

    M. Karamanlis , year = 2022, journal =. Simplices and regular polygonal tori in

    2022

  5. [5]

    Leader and P

    I. Leader and P. A. Russell and M. Walters , year = 2012, journal =. Transitive sets in

    2012

  6. [6]

    Cook and \'A

    B. Cook and \'A. Magyar and M. Pramanik , year = 2017, journal =. A

    2017

  7. [7]

    2017 , journal=

    Transitive sets and cyclic quadrilaterals , author=. 2017 , journal=

    2017

  8. [8]

    Frankl and A

    N. Frankl and A. Kupavskii and A. Sagdeev , journal=. Max-norm. 2024 , volume=

    2024

  9. [9]

    Kupavskii and A

    A. Kupavskii and A. Sagdeev , title=. Forum of Mathematics, Sigma , volume=. 2021 , pages=

    2021

  10. [10]

    J. H. Conway and A. Hulpke and J. McKay , journal=. On

  11. [11]

    Dokchitser , title =

    T. Dokchitser , title =

  12. [12]

    Ivan and I

    M.-R. Ivan and I. Leader and M. Walters , title =. Forum of Mathematics, Sigma , year =

  13. [13]

    Behague , year=

    N. Behague , year=. Nearly all known

  14. [14]

    I. K. All trapezoids are. Discrete Mathematics , year =

  15. [15]

    I. K. Permutation groups in. Proceedings of the American Mathematical Society , volume=