REVIEW 4 minor 4 references
One-point extensions of Euclidean Ramsey sets
T0 review · 0 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash
Pith's one-line read One-point extensions of finite Euclidean Ramsey sets are Ramsey.
desk verdict New proof of a theorem Moore already proved, but the proof technique is original and the paper is honest about priority. 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 load-bearing construction is the diagonal embedding $D(x)=(x,a_1,\dots,a_n)$ inside the $(n+1)$-fold product $C_n^{n+1}$, where $a_i=(1-i/n)x_0+(i/n)y$ are equally spaced points on the segment from $x_0$ to $y$. A cyclic coordinate shift $b$ acts on the product and respects the equivalence relation whose only nontrivial class is $X$. The orbit-gluing theorem then forces the class of $D(X)$ and the class of $bz_0$ to be monochromatic in every sufficiently high-dimensional colouring. The identity $\|D(x)-bz_0\|^2=\|x-y\|^2+\|y-x_0\|^2/n$ shows the added height is $\|y-x_0\|/\sqrt{n}$, which can be made arbitrarily small; the product theorem then scales this configuration up to any prescribed nonzero height.
What would settle it
A counterexample would be a finite Ramsey set $X$, a point $z\notin\mathrm{aff}(X)$, and a colouring of every $\mathbb{R}^N$ that avoids monochromatic copies of $X\cup\{z\}$; the proof's cyclic construction is explicit enough that checking it on a regular-simplex base would reveal whether such a colouring can exist.
Extended reading notes
Core claim
The central claim is Theorem 1.1: for a finite Ramsey set $X\subseteq\mathbb{R}^d$ and any point $z\notin\mathrm{aff}(X)$, the extended set $X\cup\{z\}$ is Ramsey. The proof splits into two tiers. When the orthogonal projection of $z$ lies in $\mathrm{conv}(X)$ and its height is at least the weighted radius $\rho_X(y)$, a direct diagonal product yields a monochromatic copy. The general case is handled with the language of $E$-Ramsey configurations, where an equivalence relation $E$ tells which points must share a colour. The construction places $n+1$ equally spaced points from an $x_0\in X$ to $y$, forms an $(n+1)$-fold product, uses a cyclic shift to glue the diagonal copy of $X$ to the shifted apex through the orbit-gluing theorem, and obtains an isometric copy of $X\cup\{z\}$. The height produced by the cyclic step is $\|y-x_0\|/\sqrt{n}$, which shrinks to $0$; multiplying by a two-point set then reaches any prescribed nonzero height.
Load-bearing premise
The proof's general case imports the orbit-gluing theorem as a black box, and the entire argument collapses if that theorem does not apply to the cyclic shift and equivalence relation used here.
Editorial extensions
If this is right
- Every pyramid with a finite Ramsey base is Ramsey: the apex may be any point outside the base's affine hull.
- The Ramsey class is closed under sequential one-point extensions, so any finite set grown from a Ramsey base by adding points outside the current affine span is again Ramsey.
- The proof needs no transitivity or subsolubility of the base, removing the main restriction from the earlier generalized-prism construction.
- Because the construction realizes arbitrarily small apex heights and then rescales, the size of the height is never an obstruction.
- Together with the known closure of Ramsey sets under products and scaling, this yields many new explicit Ramsey sets in high dimensions.
Reading between the lines
- The orbit-gluing template may apply beyond one-point extensions, for instance to attach several exterior points at once when the base has additional symmetries; the paper does not pursue this.
- The fact that heights tend to zero suggests the Ramsey property of such extensions is scale-invariant in the direction of the apex; whether arbitrarily small heights already suffice for every finite Ramsey base is a natural open question.
- One could test the construction computationally on small bases such as regular simplices: the equivalence classes in the product are explicit, so a direct search could reveal whether the orbit-gluing step gives tight dimension bounds.
Formalized claims in Lean
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Claim #1: The central claim is Theorem 1.1: for a finite Ramsey set $X\subseteq\mathbb{R}^d$ and any point $z\notin\mathrm{aff}(X)$, the extended set $X\cup\{z\}$ is Ramsey. The proof splits into two tiers. When the orthogonal projection of $z$ lies in $\mathrm{conv}(X)$ and its height is at least the weighted radius $\rho_X(y)$, a direct diagonal product yields a monochromatic copy. The general case is han
/-- @claim 1 The central claim is Theorem 1.1: for a finite Ramsey set $X\subseteq\mathbb{R}^d$ and any point $z\notin\mathrm{aff}(X)$, the extended set $X\cup\{z\}$ is Ramsey. The proof splits into two tiers. When the orthogonal projection of $z$ lies in $\mathrm{conv}(X)$ and its height is at least the weighted radius $\rho_X(y)$, a direct diagonal product yields a monochromatic copy. The general case is han -/ def central_claim : Prop :=
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper proves Theorem 1.1: if X is a finite Euclidean Ramsey set and z is a point outside the affine hull of X, then X∪{z} is also Ramsey, thereby answering Conjecture 8 of Ivan, Leader, and Walters. The proof proceeds in two stages. Proposition 2.1 gives an elementary diagonal-product construction under the additional assumptions that the orthogonal projection of z onto aff(X) lies in conv(X) and that the height is sufficiently large. For the general case, Section 3 uses Kříž's E-Ramsey framework: Lemma 3.1 shows that a finite set C containing X is E_C-Ramsey, where X is the only nonsingleton class; a cyclic product construction then produces one-point extensions with arbitrarily small heights, and Kříž's orbit-gluing theorem is invoked to merge the color classes of two points, yielding a monochromatic copy of S_n. A final product step scales the height to any prescribed nonzero value. The argument is self-contained except for two cited theorems of Kříž and requires no transitivity assumption on X.
Significance. If correct, the paper resolves a natural open conjecture in Euclidean Ramsey theory and significantly extends the recent generalized-prism work of Ivan, Leader, and Walters. The proof is elegant: the elementary Proposition 2.1 is a nice standalone result, and the use of E-Ramsey configurations and orbit gluing in Section 3 is a technically clean way to remove the projection and height restrictions. All algebra and isometry computations in the manuscript check out, and the dependencies on external theorems are clearly stated. The paper also acknowledges independent work by Moore. I consider the result to be a solid contribution to the field.
minor comments (4)
- [Section 2, Proposition 2.1] In the definition of ρ_X(y)^2, the quantity ρ_X(y) itself is never explicitly defined; the reader must infer from the inequality |λ| ≥ ρ_X(y) that ρ_X(y) is the square root of the right-hand side. Please make this explicit.
- [Section 3, proof of Theorem 3.2] After applying the orbit-gluing theorem, the sentence 'The relation U(E;z0,b,2) merges the E-classes of z0 and bz0' is true because U contains E and the pair (z0,bz0); spelling this out would help readers unfamiliar with the construction.
- [General] The note about the relationship to Moore's preprint is a bit unusual in the main text; it might be better placed in the acknowledgements or a footnote, although it is not problematic.
- [Section 3, Kříž's theorem citation] Please verify that the statement of Kříž's Theorem 4.1 as quoted in Section 3 indeed covers the partial-orbit gluing with arbitrary r≥1, since the proof applies it with r=2. If the theorem in [3] is only stated for full orbits, the proof still works by replacing r=2 with r=n+1; a brief remark to this effect would preempt any reader concern.
Circularity Check
No significant circularity: the proof derives the theorem from external published Ramsey-product and orbit-gluing theorems, with no fitted quantity renamed as a prediction.
full rationale
The paper's derivation chain is self-contained from stated external results. Proposition 2.1 constructs the one-point extension inside a product of scaled copies of X and a two-point set; the distance computation is explicit and the Ramsey property follows from standard closure properties, not from the conclusion. Theorem 3.2 invokes Kříž's product theorem and orbit-gluing theorem as black-box external results, applies them to configurations whose E-Ramsey property is established by Lemma 3.1, and then computes the geometry of S_n directly: the distance from D(X) to bz0 is ||x-y||^2 + ||y-x0||^2/n, giving the desired height ||y-x0||/sqrt(n). The final lifting to arbitrary height uses the Ramsey two-point set and the product theorem. No parameter is fitted to data and then called a prediction; no theorem is imported from a self-citation chain; the note about Moore's preprint is a provenance remark, not a load-bearing citation. The only significant external dependency is Kříž's orbit-gluing theorem, which is a published mathematical result cited as a theorem rather than assumed in the form of the paper's target. Hence the circularity score is 0.
Assumptions & free parameters
assumptions (3)
- standard math Standard closure properties of Euclidean Ramsey sets: invariance under nonzero scaling, inheritance by subsets, and closure under finite Cartesian products (Erdős et al. [1]).
- standard math Kříž's product theorem: if F1 is E1-Ramsey and F2 is E2-Ramsey, then F1×F2 is (E1×E2)-Ramsey; same for finite products ([3, Theorem 3.2]).
- standard math Kříž's orbit-gluing theorem: if F is E-Ramsey and b is an isometry respecting E, then F is U(E;z,b,r)-Ramsey ([3, Theorem 4.1]).
Cite this review
Pith. "Pith review of One-point extensions of Euclidean Ramsey sets." pith.science (2026). https://pith.science/paper/M3NYG2SF
@misc{pith2026260811736,
author = {Pith},
title = {Pith review of: One-point extensions of Euclidean Ramsey sets},
year = {2026},
howpublished = {\url{https://pith.science/paper/M3NYG2SF}},
note = {Machine review of arXiv:2608.11736}
}
abstract
Let $X$ be a finite Euclidean Ramsey set. We prove that adjoining any point outside the affine hull of $X$ gives another Euclidean Ramsey set, answering a conjecture of Ivan, Leader, and Walters. We first give an elementary product proof under the additional assumptions that the orthogonal projection of the new point lies in $conv(X)$ and that its distance from $aff(X)$ is sufficiently large. We then prove the general case by combining the product theorem for $E$-Ramsey configurations and K\v{r}\'{\i}\v{z}'s orbit-gluing theorem with a cyclic construction. No transitivity assumption on $X$ is needed.
Reference graph
Works this paper leans on
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[4]
A pyramid with a Ramsey base is Ramsey
K. Moore,A pyramid with a Ramsey base is Ramsey, arXiv:2608.09649 (2026). The Taft School, Watertown CT 06795, USA, and Wesleyan University, Middletown CT 06459, USA. Email: mmirabi@wesleyan.edu Website: https://sites.google.com/site/mostafamirabi/ 5
work page Pith review arXiv 2026
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[1]
P. Erd˝ os, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus, Euclidean Ramsey theorems. I, J. Combin. Theory Ser. A14(1973), 341–363
work page 1973
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[2]
M.-R. Ivan, I. Leader and M. Walters,Generalised prisms and Euclidean Ramsey theory, arXiv:2606.13472 (2026)
work page Pith review arXiv 2026
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[3]
Kˇ r´ ıˇ z,Permutation groups in Euclidean Ramsey theory, Proc
I. Kˇ r´ ıˇ z,Permutation groups in Euclidean Ramsey theory, Proc. Amer. Math. Soc.112 (1991), no. 3, 899–907
work page 1991
Reviewed August 16, 2026 · model on record in the stance chip above.
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