Monochromatic unit equilateral triangle on low-dimensional spheres

TL;DR

This paper precisely determines the dimension threshold on spheres where monochromatic unit equilateral triangles necessarily appear in 2-colorings, extending Euclidean Ramsey results to low-dimensional spheres.

Contribution

It establishes the exact dimension at which monochromatic unit equilateral triangles must occur in 2-colorings of spheres, resolving a specific case in Euclidean Ramsey theory.

Findings

No monochromatic unit equilateral triangle in 2-colorings of (1/2) sphere.

Every 2-coloring of (1/2) sphere contains a monochromatic triangle.

Determined the threshold dimension for the existence of monochromatic equilateral triangles.

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.