Diameter-Ramsey triangles below the $135^\circ$

Published 23 Apr 2026 in math.CO | (2604.22090v1)

Abstract: A finite Euclidean set is diameter-Ramsey if, for every number of colors, some finite same-diameter witness has the property that every coloring of the witness contains a monochromatic congruent copy of the set. Frankl, Pach, Reiher and Rödl asked whether any obtuse triangle is diameter-Ramsey. We prove the stronger statement that every non-degenerate triangle whose largest angle is strictly smaller than $135^\circ$ is diameter-Ramsey. Together with the theorem of Corsten and Frankl that triangles with an angle larger than $135^\circ$ are not diameter-Ramsey, this gives the sharp classification for the two open angular ranges on either side of $135^\circ$. The proof uses a weighted $k$-subset configuration with non-negative coefficients; a finite binary-tree construction realizes the required two prescribed overlaps, and the ordinary hypergraph Ramsey theorem then forces a monochromatic copy of the triangle.

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Authors (1)

Yaping Mao

Summary

The paper proves every non-degenerate triangle with a largest angle below 135° is diameter-Ramsey by constructing a finite configuration with the same diameter.
It employs weighted k-subset configurations and hypergraph Ramsey theorems to guarantee a monochromatic congruent copy under any r-coloring.
The results resolve the open obtuse triangle question and challenge conjectures based solely on the triangle's circumcenter location.

Classification of Diameter-Ramsey Triangles Below $135^\circ$

Introduction

This paper provides a sharp classification of non-degenerate triangles in the Euclidean plane with respect to the diameter-Ramsey property, focusing on those with largest angle strictly less than $135^\circ$ (2604.22090). The diameter-Ramsey property, a strengthening of classical Euclidean Ramsey theory, asks whether, for every $r$ -coloring, a same-diameter finite set can guarantee a monochromatic congruent copy of a given configuration. The result addresses an open problem concerning obtuse triangles, determining the exact angular threshold for which the diameter-Ramsey property holds.

Background

In Euclidean Ramsey theory, a finite set $P$ is Ramsey if, for all integers $r \geq 2$ , there exists a finite Euclidean set $R$ such that every $r$ -coloring of $R$ contains a monochromatic congruent copy of $P$ . Frankl and Rödl established that all triangles, and in fact all non-degenerate simplices, are Ramsey. The diameter-Ramsey property introduced by Frankl, Pach, Reiher, and Rödl further demands that the witness set $R$ has the same diameter as $135^\circ$ 0.

Prior work established necessary conditions for triangles: all acute and right triangles are diameter-Ramsey, while triangles with largest angle exceeding $135^\circ$ 1 are not. Corsten and Frankl improved this, showing that triangles with angles exceeding $135^\circ$ 2 fail the diameter-Ramsey property. However, the question remained open for obtuse triangles with largest angle strictly below $135^\circ$ 3.

Main Contribution

The central result asserts that every non-degenerate triangle with largest angle strictly less than $135^\circ$ 4 is diameter-Ramsey. This fills the gap left by previous work and provides a sharp dichotomy at the $135^\circ$ 5 threshold:

If $135^\circ$ 6 (largest angle), every triangle is diameter-Ramsey.
If $135^\circ$ 7, no triangle is diameter-Ramsey.

The argument establishes that the only cases not settled are those with largest angle exactly $135^\circ$ 8.

Methods and Techniques

The proof constructs, for an arbitrary triangle with $135^\circ$ 9, a finite Euclidean configuration with the same diameter, enforcing the monochromatic appearance of a congruent triangle under any finite coloring.

Key technical tools include:

Weighted $r$ 0-subset configurations: The construction uses vectors of the form $r$ 1 for $r$ 2-subsets $r$ 3 of $r$ 4, with non-negative weights $r$ 5. The non-negativity is essential for diameter control.
Finite binary-tree construction: A carefully crafted family of $r$ 6-subsets with prescribed pairwise overlaps is constructed. This realizes three points in the witness set whose mutual distances match the side lengths of the triangle.
Reduction to hypergraph Ramsey theorem: The $r$ 7-uniform hypergraph Ramsey number $r$ 8 ensures that, within a large enough configuration, any coloring yields a monochromatic triple realizing the distances required.
Sharp geometric translation: The condition $r$ 9 is equated to the existence of parameters $P$ 0 encoding side lengths, with the inequality $P$ 1 ensuring the feasibility of the construction.

The novelty lies in the explicit realization of the necessary Gram matrix via non-negative weights and in the mapping from angular constraints to algebraic inequalities governing the configuration.

Implications

Theoretical Implications

Resolution of the obtuse triangle question: The result proves the existence of diameter-Ramsey obtuse triangles, and fully classifies which triangles possess this property in terms of their maximal angle.
Disproof of the Corsten-Frankl simplex conjecture: The threshold contradicts a conjecture proposing that a simplex is diameter-Ramsey if and only if its circumcenter lies in its convex hull, as obtuse triangles with circumcenters outside their hull can still be diameter-Ramsey for largest angles below $P$ 2.
Sharp threshold phenomenon: The precise dichotomy at $P$ 3 demonstrates a boundary in the space of triangles separating those that have the diameter-Ramsey property from those that do not.

Practical and Future Directions

The methods extend to possible classifications of higher-dimensional simplices regarding the diameter-Ramsey property, with similar techniques potentially being fruitful for boundary cases.
The endpoint case $P$ 4 remains unresolved, as strict inequalities underlie both the constructive and the obstruction results. This suggests a discrete-to-continuum phenomenon at the sharp boundary.
The approach may inform constructions in geometric Ramsey theory and the analysis of extremal properties of Euclidean configurations under coloring.

Technical Highlights and Numerical Claims

Strong structural claim: Every triangle with largest angle strictly less than $P$ 5 is diameter-Ramsey, independent of other geometric properties such as circumcenter location.
Sharp negative result: No triangle with largest angle strictly exceeding $P$ 6 is diameter-Ramsey, due to circumradius constraints exceeding $P$ 7.
Explicit construction: For each triangle $P$ 8 with the specified angle bound and for any $P$ 9, a finite set $r \geq 2$ 0 of the same diameter explicitly guarantees a monochromatic copy of $r \geq 2$ 1.

Conclusion

This work establishes a sharp classification for the diameter-Ramsey property of non-degenerate triangles, pinpointing the exact threshold at the $r \geq 2$ 2 maximal angle. The result settles a longstanding open problem and demonstrates that circumcenter location is insufficient for characterizing diameter-Ramsey simplices in two dimensions. Future work may revisit the $r \geq 2$ 3 endpoint and explore analogous threshold phenomena in higher-dimensional configurations.