Monochromatic unit equilateral triangle on low-dimensional spheres

Abstract

A result of Matoušek and Rödl in 1995 states that for every $\varepsilon>0$ and every triangle $T$ with circumradius $ρ(T)$, there exists a dimension $n=n(\varepsilon,T)$ such that every $2$-coloring of the $n$-dimensional sphere of radius $ρ(T)+\varepsilon$, namely $\mathbb{S}^{n}(ρ(T)+\varepsilon)$, 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.