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arxiv: 2605.16958 · v1 · pith:WBSKRL5Ynew · submitted 2026-05-16 · 🧮 math.CO · math.MG

Monochromatic unit equilateral triangle on low-dimensional spheres

Xiaochen Zhao , Gennian Ge This is my paper

Pith reviewed 2026-05-19 20:15 UTC · model grok-4.3

classification 🧮 math.CO math.MG
keywords monochromatic trianglessphere coloringsRamsey theoryequilateral triangleslow-dimensional spheresEuclidean Ramsey problems2-colorings of spheres
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The pith

There exists a 2-coloring of the 2-sphere of radius 1/√2 with no monochromatic unit equilateral triangle, but every 2-coloring of the 3-sphere contains one.

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

The paper determines the precise dimension at which monochromatic copies of the unit equilateral triangle become unavoidable under 2-colorings of spheres with radius 1/√2. It constructs an explicit coloring that avoids such triangles on the two-dimensional sphere and proves that any coloring of the three-dimensional sphere must include one. This sharpens a 1995 result showing that monochromatic copies appear in all sufficiently high dimensions for any fixed triangle. A reader cares because the work identifies the exact low-dimensional threshold for this geometric configuration rather than an asymptotic existence statement.

Core claim

The authors prove that there exists a 2-coloring of S²(1/√2) containing no monochromatic unit equilateral triangle defined by chord distance 1, whereas every 2-coloring of S³(1/√2) contains at least one such triangle. They also obtain related results for asymmetric and isosceles triangles on these low-dimensional spheres.

What carries the argument

The unit equilateral triangle on the sphere, with sides given by Euclidean chord distance exactly 1 in the ambient space.

If this is right

  • The threshold dimension for forcing a monochromatic unit equilateral triangle is exactly 3 for spheres of this radius.
  • The same low-dimensional spheres admit colorings or forced configurations for certain asymmetric and isosceles triangles as well.
  • The general high-dimensional existence result of Matoušek and Rödl is realized already at dimension 3 for this particular triangle and radius.

Where Pith is reading between the lines

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

  • Similar exact thresholds might exist for other fixed distances or other regular polygons on the same spheres.
  • The explicit coloring avoiding the triangle on the 2-sphere could be used to test related questions about larger sets or higher chromatic numbers.
  • The geometric choice of radius 1/√2 aligns the target chord distance with right angles in the ambient coordinates, which may simplify further calculations.

Load-bearing premise

Distances are measured by straight-line chords through the ambient Euclidean space, and the sphere radius is fixed so that chord distance 1 is a geometrically natural length on that sphere.

What would settle it

An explicit 2-coloring of the three-dimensional sphere of radius 1/√2 that contains no three points at mutual chord distance 1 forming a monochromatic triangle would falsify the claim that every such coloring contains one.

read the original abstract

A result of Matou\v{s}ek and R\"odl in 1995 states that for every $\varepsilon>0$ and every triangle $T$ with circumradius $\rho(T)$, there exists a dimension $n=n(\varepsilon,T)$ such that every $2$-coloring of the $n$-dimensional sphere of radius $\rho(T)+\varepsilon$, namely $\mathbb{S}^{n}(\rho(T)+\varepsilon)$, contains a monochromatic congruent copy of $T$. In this paper, we determine the exact threshold dimension for the unit equilateral triangle on the sphere $\mathbb{S}^{n}(1/\sqrt{2})$: there exists a $2$-coloring of $\mathbb{S}^{2}(1/\sqrt{2})$ with no monochromatic unit equilateral triangle, whereas every $2$-coloring of $\mathbb{S}^{3}(1/\sqrt{2})$ contains one. Along the way, we also establish several further Euclidean Ramsey-type results on low-dimensional spheres, including asymmetric and isosceles variants.

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

1 major / 3 minor

Summary. The manuscript determines the exact threshold dimension for the monochromatic unit equilateral triangle in 2-colorings of spheres of radius 1/√2. It constructs an explicit 2-coloring of S^2(1/√2) containing no monochromatic unit equilateral triangle and proves that every 2-coloring of S^3(1/√2) contains one. Additional results on asymmetric and isosceles variants are established along the way.

Significance. If the central claims hold, the work supplies a sharp low-dimensional threshold that refines the general high-dimensional existence theorem of Matoušek and Rödl (1995). The concrete construction on the 2-sphere and the forcing argument on the 3-sphere furnish explicit examples that clarify the transition point in Euclidean Ramsey theory for this configuration.

major comments (1)
  1. [Section 4] The proof that every 2-coloring of S^3(1/√2) contains a monochromatic unit equilateral triangle (appearing in the section establishing the positive result for dimension 3) must explicitly address whether the argument applies to arbitrary colorings or requires the color classes to be Lebesgue measurable (or closed). Without such clarification, the universal quantifier is vulnerable to axiom-of-choice counterexamples, which is load-bearing for the main theorem.
minor comments (3)
  1. [Section 2] The definition of the unit equilateral triangle via Euclidean chord distance should be restated with an explicit formula in the preliminaries to avoid any ambiguity with spherical distance.
  2. [Introduction] The citation to Matoušek and Rödl should be expanded to include the full bibliographic details and page range.
  3. [Section 3] Figure 2 illustrating the coloring of S^2(1/√2) would benefit from an additional legend distinguishing the two color classes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comment on the scope of the main positive result. We address the point directly below and will incorporate a clarification in the revised version.

read point-by-point responses

  1. Referee: [Section 4] The proof that every 2-coloring of S^3(1/√2) contains a monochromatic unit equilateral triangle (appearing in the section establishing the positive result for dimension 3) must explicitly address whether the argument applies to arbitrary colorings or requires the color classes to be Lebesgue measurable (or closed). Without such clarification, the universal quantifier is vulnerable to axiom-of-choice counterexamples, which is load-bearing for the main theorem.

    Authors: We appreciate the referee drawing attention to this subtlety. The argument in Section 4 applies to arbitrary 2-colorings. It proceeds by contradiction: assume a 2-coloring of S^3(1/√2) with no monochromatic unit equilateral triangle; the geometry of the 3-sphere then forces a finite collection of points whose color patterns are impossible by a direct case analysis that uses only the metric properties of the sphere and the definition of unit equilateral triangles. No integration, averaging, or measurability is invoked, so the reasoning holds for every function from the sphere to {red, blue} and does not admit axiom-of-choice counterexamples. In the revised manuscript we will insert a short paragraph at the start of Section 4 making this explicit and stating that the result is claimed for arbitrary colorings. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit construction and proof are independent of inputs

full rationale

The paper states an explicit 2-coloring avoiding monochromatic unit equilateral triangles on S^2(1/√2) and proves that every 2-coloring of S^3(1/√2) contains one, building directly on the 1995 Matoušek-Rödl result without reducing any claim to a fitted parameter, self-definition, or self-citation chain. The derivation relies on combinatorial partitions and case analysis on orthogonal frames rather than any equation or premise that loops back to the target statement by construction. No load-bearing ansatz, uniqueness theorem from the same authors, or renaming of known results appears in the abstract or described chain; the result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no free parameters, invented entities, or non-standard axioms beyond the usual properties of Euclidean spheres and distance.

axioms (1)
  • standard math Standard Euclidean geometry on spheres embedded in R^{n+1}
    The definitions of sphere, chord distance, and equilateral triangle rely on classical properties of Euclidean space.

pith-pipeline@v0.9.0 · 5703 in / 1134 out tokens · 47421 ms · 2026-05-19T20:15:58.479495+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    there exists a 2-coloring of S²(1/√2) with no monochromatic unit equilateral triangle, whereas every 2-coloring of S³(1/√2) contains one

  • Foundation/AlexanderDuality.lean D3_admits_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    a unit equilateral triangle on S is exactly a triple {d1,d2,d3} ⊆ S such that di · dj = 0 for all i≠j

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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