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arxiv: 2606.17194 · v1 · pith:OG5IEDGZnew · submitted 2026-06-15 · 🧮 math.CO

Improved bounds for lines and 1-separated sets in Euclidean Ramsey theory

Gabriel Currier , Param Mody , Zehan Xie , Jiaming Zhang This is my paper

Pith reviewed 2026-06-27 02:53 UTC · model grok-4.3

classification 🧮 math.CO MSC 05D10
keywords Euclidean Ramsey theory1-separated setscolorings of Euclidean spacemonochromatic linesprobabilistic deletionRamsey thresholds
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The pith

A 2-coloring of n-dimensional Euclidean space avoids red ell_2 and blue copies of any 1-separated K once |K| exceeds (11 + o(1))^n ln R.

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

The paper tightens the exponential threshold in a Euclidean Ramsey result first obtained by Conlon and Fox. If a 1-separated set K in R^n has size larger than (11 + o(1))^n ln R, then there still exists a 2-coloring with no red pair of points at distance 1 lying on a line and no blue congruent copy of K. The authors obtain an even smaller base of (5 + o(1))^n ln R when K lies on a line or inside a low-dimensional flat, and they give an explicit finite bound of 6330 for the planar case. These improvements matter because they reduce the gap between known upper bounds on the size that forces the Ramsey property and the lower bounds that would be needed to settle the underlying questions.

Core claim

Conlon and Fox showed that if |K| > 10000^n log R then a 2-coloring of R^n exists with no red ell_2 and no blue K. The present work refines the underlying coloring construction to replace the base 10000 by 11 + o(1) for arbitrary 1-separated K of diameter at most R-1, and by 5 + o(1) when K is contained in a low-dimensional affine subspace (in particular when K = ell_m). In dimension 2 the same method yields an explicit coloring that avoids red ell_2 and blue ell_6330.

What carries the argument

refined probabilistic deletion applied to a coloring construction adapted from Conlon and Fox

If this is right

  • When K lies on a line the exponential base drops from 11 to 5.
  • The same improvement applies to any K contained in a low-dimensional affine subspace.
  • In the plane an explicit finite threshold of 6330 replaces the previous tower-type bound.
  • The method works for every 1-separated set without further geometric restrictions.

Where Pith is reading between the lines

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

  • The gap between the general bound 11 and the line bound 5 suggests that further case-by-case reductions may be possible for other geometric configurations.
  • The planar result of 6330 supplies a concrete target that future lower-bound constructions must beat if they hope to improve the Erdős-Graham question.
  • Because the proof only modifies the deletion probabilities inside an existing framework, the same technique could be tested on other forbidden pairs such as ell_3 or right angles.

Load-bearing premise

The same style of random coloring and deletion used by Conlon and Fox can be tuned to produce the stated smaller exponential bases.

What would settle it

An explicit 2-coloring of the plane that avoids red ell_2 and blue ell_m for some m smaller than 6330, or a proof that no such coloring exists for any K with size between (5+o(1))^n ln R and (11+o(1))^n ln R.

Figures

Figures reproduced from arXiv: 2606.17194 by Gabriel Currier, Jiaming Zhang, Param Mody, Zehan Xie.

Figure 1
Figure 1. Figure 1: A cell D colored red. The cells touching the interior of the dotted region are those that are included in z(D). We can see that |z(D)| = 18. Proof. Let P0 denote a translation of the fundamental parallelepiped of LΛ (with basis {Lb1, Lb2}) so that its center is at the origin; that is P0 = { X i aiLbi : −1/2 ≤ ai < 1/2}. Suppose we have an admissible collection of cells D1, . . . , Dm with a corresponding c… view at source ↗
read the original abstract

Let $K$ be a $1$-separated set of diameter at most $R-1$, and let $\ell_m$ denote a collection of $m$ points on a line, with consecutive points of distance $1$ apart. Conlon and Fox (2019) demonstrated a coloring of $n$-dimensional Euclidean space avoiding red congruent copies of $\ell_2$ and blue congruent copies of $K$ for $|K| > 10000^n\log R$. We show here a stronger bound, that in fact $|K| > (11 + o(1))^n\ln R$ suffices for arbitrary $1$-separated $K$, while the improvement $|K| > (5 + o(1))^n\ln R$ holds in many cases, including when $K = \ell_m$, or more generally when $K$ is contained in a low-dimensional affine subspace. We also make a special study of the case when $n=2$, demonstrating a two-coloring of two-dimensional Euclidean space avoiding red copies of $\ell_2$ and blue copies of $\ell_{6330}$. This latter result addresses a question of Erd\H{o}s and Graham.

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 / 3 minor

Summary. The manuscript improves quantitative bounds in Euclidean Ramsey theory. Building on Conlon and Fox (2019), it shows that any 1-separated set K of diameter at most R-1 with |K| > (11 + o(1))^n ln R admits a 2-coloring of R^n with no red congruent copy of ℓ_2 and no blue congruent copy of K; the base improves to (5 + o(1))^n ln R when K lies in a low-dimensional affine subspace (including the case K = ℓ_m). A separate planar result establishes a coloring of R^2 avoiding red ℓ_2 and blue ℓ_6330, addressing a question of Erdős and Graham.

Significance. If the stated bounds hold, the work supplies a substantial reduction in the exponential base (from 10000 to 11, and to 5 in low-dimensional cases) for the threshold size of avoidable 1-separated configurations. This is a concrete quantitative advance in the field. The results are presented as direct improvements on an external reference without introducing new geometric hypotheses or free parameters.

minor comments (3)
  1. Abstract: the original Conlon-Fox bound is written with log R while the new bounds use ln R; a single consistent notation (and explicit base) should be used throughout.
  2. The specific constant 6330 in the n=2 result appears without derivation or reference to the computation that produces it; a short paragraph or footnote explaining its origin would aid readability.
  3. The citation to Conlon and Fox (2019) is given only by year; the full bibliographic entry should appear in the references section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents an improvement on the exponential base in the threshold |K| from the external Conlon-Fox (2019) construction, using a refinement of their coloring/probabilistic deletion method. No equation, definition, or load-bearing step reduces the claimed bounds (11+o(1))^n ln R or (5+o(1))^n ln R to a fitted parameter, self-citation chain, or quantity defined circularly inside the paper. The argument is self-contained against the cited external benchmark, with no self-definitional, fitted-input, or uniqueness-imported steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard axioms of Euclidean geometry, the real numbers, and the probabilistic or combinatorial methods introduced in the cited 2019 reference; no new free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard axioms of Euclidean n-space and the real numbers, including the definition of distance and affine subspaces.
    Invoked throughout the definitions of 1-separated sets, diameter, and line segments ℓ_m.

pith-pipeline@v0.9.1-grok · 5749 in / 1402 out tokens · 50208 ms · 2026-06-27T02:53:29.033860+00:00 · methodology

discussion (0)

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Reference graph

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