Monochromatic Equilateral Triangles in the Unit Distance Graph

Abstract

Let $\chi_{\Delta}(\mathbb{R}^{n})$ denote the minimum number of colors needed to color $\mathbb{R}^{n}$ so that there will not be a monochromatic equilateral triangle with side length $1$. Using the slice rank method, we reprove a result of Frankl and Rodl, and show that $\chi_{\Delta}\left(\mathbb{R}^{n}\right)$ grows exponentially with $n$. This technique substantially improves upon the best known quantitative lower bounds for $\chi_{\Delta}\left(\mathbb{R}^{n}\right)$, and we obtain \[ \chi_{\Delta}\left(\mathbb{R}^{n}\right)>(1.01446+o(1))^{n}. \]