%A Sagdeev, Arsenii A.
%A Raigorodskii, Andrei M.
%D 2019
%T On a Frankl-Wilson theorem and its geometric corollaries
%K
%X 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 $(\xi_m+o(1))^n$.
%U http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1216
%J Acta Mathematica Universitatis Comenianae
%0 Journal Article
%P 1029-1033%V 88
%N 3
%@ 0862-9544
%8 2019-07-31