On a Frankl-Wilson theorem and its geometric corollaries

Abstract

We find a new analogue of the Frankl--Wilson theorem on the independence number of distance graphs of some special type. We apply this new result to two combinatorial geometry problems.First, we improve a previously known value $c$ such that $\chi\left( \mathbb{R}^n; S_2\right) \geq \left(c+o(1)\right)^n$, where $\chi\left( \mathbb{R}^n; S_2\right)$ is the minimum number of colors needed to color all points of $\mathbb{R}^n$ so that there is no monochromatic set of vertices of a unit equilateral triangle $S_2$.Second, given $m \geq 3$ we improve the value $\xi_m$ such that for any $n \in \mathbb{N}$ there is a distance graph in $\mathbb{R}^n$ with the girth greater than $m$ and the chromatic number at least $(\xi_m+o(1))^n$.