REVIEW 5 minor 2 cited by
A pyramid with a Ramsey base is Ramsey
T0 review · 0 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read A finite Ramsey set remains Ramsey after adding any point outside its affine hull.
desk verdict A short, clean proof of a conjecture from [7]: any one-point extension of a Ramsey set outside its affine hull is Ramsey; the argument is sound and worth publishing. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The construction centres on three objects: the finite forcing configuration C supplied by the compactness lemma, the simplex S formed by adding a fixed perpendicular height h to each point of C, and the product P = B × S. The distance identity ||(b,0) − z||² = ||b − u||² + h² makes each fibre Pi ∪ {a_i} congruent to X, and this identity is the pivot that turns a monochromatic copy of P into a monochromatic copy of X. The product theorem and the simplex theorem guarantee that P is Ramsey, while the compactness lemma ensures that the auxiliary configuration C can be chosen finite.
What would settle it
A concrete counterexample—a finite Ramsey set B and a point z outside its affine hull such that B∪{z} is not Ramsey for some number of colours—would refute the theorem; testing small bases such as a segment or a square for two colours is the natural place to look.
Extended reading notes
Core claim
Theorem 1.2: for any finite Ramsey set B and any vector z outside the affine hull of B, the one-point extension B∪{z} is Ramsey. The proof establishes this by an inductive colour-count argument: assuming the pyramid is Ramsey for r−1 colours, one uses a finite-forcing lemma to obtain a finite configuration C that already r−1-forces X, then builds a Ramsey product P of B with an affinely independent simplex S whose vertices sit 'above' the points of C at a fixed height h. Each fibre of P together with the corresponding apex is congruent to X, so in any r-colouring a monochromatic copy of P yields either a red apex, completing X, or a fully non-red copy of C, which by induction contains X.
Load-bearing premise
The proof requires that whenever a dimension forces a monochromatic copy of the target set with r−1 colours, some finite subset of that dimension already does; without this compactness property, the finite configuration on which the entire product-simplex construction is built would not exist.
Editorial extensions
If this is right
- Any finite set built by repeatedly adding points outside the current affine hull of a Ramsey base remains Ramsey, since the theorem can be iterated.
- The theorem gives a pyramid construction that does not require the base to be transitive or subsoluble; Ramsey-ness of the base alone suffices.
- The result is entailed by either of the two competing classification conjectures for Ramsey sets, so it is consistent with both and does not discriminate between them.
- Together with the known fact that all Ramsey sets are spherical, the theorem shows that the class of Ramsey sets is closed under this kind of one-point lifting, so any counterexample to the classification must arise in some other way.
Reading between the lines
- The proof's mechanism is scale-invariant in the perpendicular direction: for any nonzero height h, the same combinatorial argument produces a Ramsey pyramid, so the result holds for every point sharing the same orthogonal projection onto the affine hull as z.
- Because the added point z need not carry any symmetry relative to B, the theorem suggests that Ramsey-ness may be much more robust than symmetry-based classifications imply; whether iterating such asymmetric extensions eventually forces subtransitivity or sphericity is a natural next question.
- The finite-forcing lemma is the only non-elementary ingredient besides the two classical theorems; a constructive or quantitative version of that lemma could turn the proof into an explicit bound on the Ramsey dimension of pyramids.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves Theorem 1.2: if B is a finite Ramsey set in a Euclidean space and z is a point outside the affine hull of B, then B∪{z} is also Ramsey. This answers Conjecture 8 of Ivan, Leader, and Walters. The proof is by induction on the number of colors r. For r=1 the claim is trivial. For r≥2, assuming X=B∪{z} is Ramsey for r−1 colors, the author uses a compactness consequence of the de Bruijn–Erdős theorem to obtain a finite configuration C such that every (r−1)-coloring of C contains a monochromatic copy of X. He then builds a simplex S in a higher-dimensional space whose vertices are the points of C lifted by orthogonal basis vectors, and takes the Cartesian product P=B×S. By the product theorem and the Frankl–Rödl simplex theorem, P is Ramsey. In any r-coloring of a sufficiently large Euclidean space, a monochromatic copy of P appears. Depending on the colors of the images of the corresponding apex points, one either finds a red copy of X inside a fibre together with a red apex, or an (r−1)-colored copy of C that already contains a monochromatic copy of X. The distance computations in equations (2)–(4) are exact, and the two-case split is exhaustive.
Significance. The main result is a positive answer to a conjecture in a recent paper by Ivan, Leader, and Walters, and it adds a new closure property to the known structural results about Ramsey sets. The proof is concise and relies only on classical theorems: the product theorem of Erdős et al., the Frankl–Rödl simplex theorem, and the de Bruijn–Erdős compactness theorem. The argument is transparent and the geometry is elementary. This is a solid contribution to Euclidean Ramsey theory, even though it does not by itself distinguish between the two conjectured classifications.
minor comments (5)
- [§3, base case] The statement 'The case when r=1 is not difficult' could be made explicit: for r=1, any coloring of a space of dimension at least the dimension of X contains X as a monochromatic copy.
- [§2, Lemma 2.3] The compactness argument behind Lemma 2.3 is standard but omitted; a one-sentence indication (e.g., appealing to the de Bruijn–Erdős theorem for the hypergraph whose vertices are points of R^n and whose edges are copies of T) would improve self-containment.
- [§1, Conjecture 1.1] The phrase 'We give two competing statements' is informal; consider phrasing such as 'Two natural conjectures have been proposed.'
- [§1, paragraph 2] The term 'solubility hypothesis' is used without definition; since this is not a standard term in Euclidean Ramsey theory outside the cited paper, please clarify or cite the definition.
- [§3, notation] The arrow notation 'R^{d2} r−1−−→X' is defined but typeset awkwardly; consider using a standard \xrightarrow{r-1} in the typeset version.
Circularity Check
No significant circularity: the proof is a self-contained induction on the number of colours using classical theorems as external inputs.
full rationale
The paper does not derive its conclusion from itself. Theorem 1.2 is proved by induction on the number r of colours: the base r=1 is trivial, and the inductive step assumes only that X=B∪{z} is Ramsey for r−1 colours, which is not the statement being proved at r. That hypothesis is used to obtain, via Lemma 2.3 (a standard de Bruijn–Erdős compactness consequence from [1] and Proposition 4 in [3]), a finite configuration C with C r−1→X. The construction of the simplex S, the product P=B×S, and the apex set A is geometric and relies on the cited product theorem [3] and simplex theorem [6], both external benchmarks with independent proofs. The distance identity (3) equating with (2) is a direct computation, and the final dichotomy (red apex or r−1-coloured copy of C) exhausts all cases without assuming the target property for r colours. The only role of [7] is to pose the question and to note that the theorem is weaker than either conjecture; the proof does not use any result from [7] as a load-bearing input, and there is no self-citation chain. No fitted parameter is renamed as a prediction, and no equation is equivalent by construction to the target statement. Hence the circularity score is 0.
Assumptions & free parameters
assumptions (4)
- standard math Product theorem: if A and B are Ramsey configurations, then A×B is Ramsey.
- standard math Frankl-Rödl simplex theorem: every finite affinely independent Euclidean configuration is Ramsey.
- standard math de Bruijn-Erdős compactness lemma: if R^n r→T then some finite A⊆R^n satisfies A r→T.
- standard math Every congruence between finite subsets of R^N extends to an isometry of R^N.
Cite this review
Pith. "Pith review of A pyramid with a Ramsey base is Ramsey." pith.science (2026). https://pith.science/paper/MTN74UFP
@misc{pith2026260809649,
author = {Pith},
title = {Pith review of: A pyramid with a Ramsey base is Ramsey},
year = {2026},
howpublished = {\url{https://pith.science/paper/MTN74UFP}},
note = {Machine review of arXiv:2608.09649}
}
abstract
A finite subset $X$ of ${\mathbb R}^d$ is called a Ramsey set if for any number of colours $k$ there exists a dimension $n$ such that whenever ${\mathbb R}^n$ is $k$-coloured there exists a monochromatic congruent copy of $X$. The classification of Ramsey sets is one of the major unsolved problems in the field of Euclidean Ramsey theory. Towards this, Ivan, Leader and Walters recently asked whether adding a point to a Ramsey set outside of its affine hull necessarily produces another Ramsey set. In this note, we answer their question in the affirmative.
Figures
Forward citations
Cited by 2 Pith papers
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A nearcircumsphere-Ramsey Theorem for Solvable Transitive Configurations
Every solvable transitive finite spherical set is nearcircumsphere-Ramsey, meaning monochromatic congruent copies exist on high-dimensional spheres of radius only slightly exceeding the circumradius.
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One-point extensions of Euclidean Ramsey sets
Any finite Euclidean Ramsey set remains Ramsey after adjoining any point outside its affine hull.
Reference graph
Works this paper leans on
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[7]
M.-R. Ivan, I. Leader, and M. Walters. Generalised prisms and Euclidean Ramsey theory, 2026
work page 2026
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[1]
N. G. de Bruijn and P. Erdős. A colour problem for infinite graphs and a problem in the theory of relations.Nederl. Akad. Wetensch. Proc. Ser. A, 54:371–373, 1951. Also published in Indagationes Mathematicae 13 (1951), 369–373
work page 1951
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[4]
P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus. Euclidean Ramsey theorems. II. In A. Hajnal, R. Rado, and V. T. Sós, editors,Infinite and Finite Sets, volume 10 ofColloquia Mathematica Societatis János Bolyai, pages 529–557. North-Holland, Amsterdam, 1975. Proceedings of the Colloquium held in Keszthely, 1973
work page 1975
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[5]
P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus. Euclidean Ramsey theorems. III. In A. Hajnal, R. Rado, and V. T. Sós, editors,Infinite and Finite Sets, volume 10 ofColloquia Mathematica Societatis János Bolyai, pages 559–583. North-Holland, Amsterdam, 1975. Proceedings of the Colloquium held in Keszthely, 1973
work page 1975
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[6]
P. Frankl and V. Rödl. A partition property of simplices in Euclidean space.Journal of the American Mathematical Society, 3(1):1–7, 1990
work page 1990
- [8]
Show all 9 references
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[9]
Leader, P
I. Leader, P. A. Russell, and M. Walters. Transitive sets in Euclidean Ramsey theory. Journal of Combinatorial Theory, Series A, 119(2):382–396, 2012. 5
2012
Reviewed August 11, 2026 · model on record in the stance chip above.
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