Monochromatic Equilateral Triangles in the Unit Distance Graph

TL;DR

This paper demonstrates that the minimum number of colors needed to avoid monochromatic equilateral triangles in Euclidean space grows exponentially with dimension, using the slice rank method to improve lower bounds.

Contribution

The paper applies the slice rank method to establish exponential lower bounds for the chromatic number related to equilateral triangles in high-dimensional spaces, improving previous results.

Findings

Established exponential growth of hi_elta() with dimension n.

Improved quantitative lower bounds for hi_elta() using the slice rank method.

Proved hi_elta() > (1.01446+o(1))^n.

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