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 \[ χ_Δ\left(\mathbb{R}^{n}\right)>(1.01446+o(1))^{n}. \]

4 pages

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