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arxiv: 2604.15466 · v1 · submitted 2026-04-16 · 🧮 math.CO

Any 2-coloring of the plane contains monochromatic unit rhombuses

Kenneth Moore , Arsenii Sagdeev This is my paper

Pith reviewed 2026-05-10 10:22 UTC · model grok-4.3

classification 🧮 math.CO
keywords 2-coloring of the planemonochromatic rhombusunit distancegeometric Ramsey theoryunavoidable configurationsplane coloring
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The pith

Any 2-coloring of the plane contains four monochromatic points forming a rhombus with unit sides and non-unit diagonals.

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

The paper proves that coloring every point of the Euclidean plane red or blue must produce four points of one color that form a rhombus whose four sides each have length 1 while its diagonals have length different from 1. This settles an explicit open question left by earlier authors on whether such a configuration is forced. A sympathetic reader cares because the result belongs to the study of unavoidable geometric patterns under finite colorings, showing that even simple quadrilaterals cannot be avoided when only two colors are available. The argument works by exhibiting a finite auxiliary set of points whose every 2-coloring already contains the desired rhombus. If the claim holds, then certain distance-1 rhombi join the growing list of monochromatic structures guaranteed in the plane.

Core claim

We prove that any 2-coloring of the plane contains 4 points of the same color forming a rhombus with unit sides and non-unit diagonals, answering a question of Axenovich, Liu, and the second author.

What carries the argument

A finite point configuration in the plane that forces a monochromatic unit rhombus (sides length 1, diagonals unequal to 1) under any 2-coloring.

Load-bearing premise

The result assumes the standard Euclidean plane with its usual distance metric and that the 2-coloring partitions every point of R^2.

What would settle it

An explicit 2-coloring of all points in the plane with no four monochromatic points forming a rhombus of side length 1 whose diagonals both differ from length 1.

Figures

Figures reproduced from arXiv: 2604.15466 by Arsenii Sagdeev, Kenneth Moore.

Figure 1
Figure 1. Figure 1: The set B7, seven points with no valid coloring We prove the first step with Lemma 2 below, and because of the geometric connection from Lemma 1, this step can be phrased purely in terms of graphs. For graphs G and H, we define the Cartesian product G□H as the graph with vertex set V (G) × V (H), where (v1, v2), (u1, u2) form an edge in G□H if and only if either v1u1 ∈ E(G) and v2 = u2, or v1 = u1 and v2u2… view at source ↗
Figure 2
Figure 2. Figure 2: Two sets combined with the A ⊕3 B notation Lemma 3. Let T be the triangle with sides 1, 1, 1 6 (3 + √ 33), and let A ⊂ R 2 be a finite set. There are finite sets B154 and B46 such that (a) If A 2−→ {T} ∪ C3 ∪ C4, then A ⊕3 B154 2−→ C3 ∪ C4. (b) If any 2-coloring of A contains either two points at distance √ 4 3 that receive opposite colors or a monochromatic element of C4, then A ⊕2 B46 2−→ {T} ∪ C3 ∪ C4. … view at source ↗
Figure 3
Figure 3. Figure 3: The constructions B154 and B46 The proof of this is computer assisted. We created a resources webpage [9], where we keep text files containing coordinates of B154 and B46 (which can also be found in Appendix A), and a simple Python program. The Python program first converts a specified point set into a hypergraph, where the vertices corresponding to a set of points are connected by an edge if they geometri… view at source ↗
Figure 4
Figure 4. Figure 4: Construction of a good seed with 258 points [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

In this note, we prove that any 2-coloring of the plane contains 4 points of the same color forming a rhombus with unit sides and non-unit diagonals, answering a question of Axenovich, Liu, and the second author.

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 / 2 minor

Summary. The paper proves that any 2-coloring of the Euclidean plane contains four points of the same color that form a rhombus with all sides of length 1 and both diagonals of length different from 1. The argument proceeds by exhibiting an explicit finite point set in R^2 and exhaustively analyzing all 2-colorings of its critical points to force the desired monochromatic rhombus, relying only on the Euclidean metric and the partition property of the coloring.

Significance. If correct, the result resolves a specific open question posed by Axenovich, Liu, and the second author. It demonstrates that a non-regular geometric configuration is unavoidable in 2-colorings of the plane, contributing to the study of monochromatic structures in combinatorial geometry. The finite-configuration plus exhaustive case analysis method is a standard, self-contained technique that yields a checkable proof without measurability or continuity assumptions.

minor comments (2)
  1. [Introduction] The introduction could briefly recall the exact statement of the question from Axenovich, Liu, and the second author to make the motivation self-contained.
  2. A diagram labeling the critical points of the finite configuration and indicating which distances are 1 would improve readability of the case analysis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive summary and significance assessment, and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; direct finite-configuration proof

full rationale

The manuscript establishes the claim by exhibiting an explicit finite point set in R^2 and performing exhaustive case analysis on its 2-colorings. Each case directly invokes only the Euclidean distance and the partition property of the coloring to force a monochromatic unit-side rhombus with non-unit diagonals. No parameter is fitted, no ansatz is imported via self-citation, and the central argument does not reduce to any prior result by the same authors; the cited open question merely motivates the problem without supplying any load-bearing step of the proof. The derivation is therefore self-contained and independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the axioms of Euclidean geometry and the definition of a 2-coloring of the plane; no free parameters or invented entities are introduced in the abstract statement.

axioms (2)
  • standard math The Euclidean plane R^2 with standard distance metric
    Invoked implicitly as the space being colored.
  • domain assumption A 2-coloring is a partition of all points into two sets
    Standard assumption for Ramsey-type coloring problems.

pith-pipeline@v0.9.0 · 5322 in / 1155 out tokens · 35498 ms · 2026-05-10T10:22:01.714207+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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    Axenovich, D

    M. Axenovich, D. Liu, and A. Sagdeev. Ramsey problems for graphs in Euclidean spaces and Cartesian powers.arXiv preprint arXiv:2512.15516, 2025.↑1, 2, 6

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    Erd˝ os, R

    P. Erd˝ os, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus. Euclidean Ramsey theorems. II. InInfinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erd˝ os on his 60th birthday), Vols. I, II, III, volume Vol. 10 ofColloq. Math. Soc. J´ anos Bolyai, pages 529–557. North-Holland, Amsterdam-London, 1975.↑1

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    Erd˝ os, R

    P. Erd˝ os, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus. Euclidean Ramsey theorems. III. InInfinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erd˝ os on his 60th birthday), Vols. I, II, III, volume Vol. 10 ofColloq. Math. Soc. J´ anos Bolyai, pages 559–583. North-Holland, Amsterdam-London, 1975.↑1

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    work page 2018

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    Moore and A

    K. Moore and A. Sagdeev. Resources for ‘any 2-coloring of the plane contains monochromatic unit rhombuses’, 2026.https://users.renyi.hu/ ~kjmoore/rhombusresources.html.↑4 6 A Point coordinates The tuples representing the points ofB 154: [−1,3,−9,1],[5,3,−3,1],[−1,1,1,1],[−11,1,1,−1],[0,4,−8,0],[−6,2,−4,0],[−5,3,−3,−1], [−7,3,−3,1],[0,2,−10,0],[−1,3,3,1],[...

    work page 2026