Monochromatic unit equilateral triangle on low-dimensional spheres

Abstract

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

24 pages