Monochromatic Equilateral Triangles in the Unit Distance Graph

Let ${\chi }_{\mathrm{\Delta }}({\mathbb{R}}^n)$ 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 ${\chi }_{\mathrm{\Delta }}\left({\mathbb{R}}^n\right)$ grows exponentially with $n$. This technique substantially improves upon the best known quantitative lower bounds for ${\chi }_{\mathrm{\Delta }}\left({\mathbb{R}}^n\right)$, and we obtain $${\chi }_{\mathrm{\Delta }}\left({\mathbb{R}}^n\right)>(1.01446+o(1){)}^n\mathrm{.}$$