1Laboratory of Advanced Combinatorics and Network Applications, Moscow Institute of Physics and Technology (State University), Moscow, Russian Federation 2Laboratory of Advanced Combinatorics and Network Applications, Moscow Institute of Physics and Technology (State University); Mechanics and Mathematics Faculty, Moscow State University; Institute of Mathematics and Computer Science, Buryat State University; Caucasus Mathematical Centre, Adyghe State University, Russian Federation
Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 1029-1033
Citer cet article
Arsenii A. Sagdeev; Andrei M. Raigorodskii; Arsenii A. Sagdeev; Andrei M. Raigorodskii. On a Frankl-Wilson theorem and its geometric corollaries. Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 1029-1033. http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a104/
@article{AMUC_2019_88_3_a104,
author = {Arsenii A. Sagdeev and Andrei M. Raigorodskii and Arsenii A. Sagdeev and Andrei M. Raigorodskii},
title = { On a {Frankl-Wilson} theorem and its geometric corollaries},
journal = {Acta mathematica Universitatis Comenianae},
pages = {1029--1033},
year = {2019},
volume = {88},
number = {3},
url = {http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a104/}
}
TY - JOUR
AU - Arsenii A. Sagdeev
AU - Andrei M. Raigorodskii
AU - Arsenii A. Sagdeev
AU - Andrei M. Raigorodskii
TI - On a Frankl-Wilson theorem and its geometric corollaries
JO - Acta mathematica Universitatis Comenianae
PY - 2019
SP - 1029
EP - 1033
VL - 88
IS - 3
UR - http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a104/
ID - AMUC_2019_88_3_a104
ER -
%0 Journal Article
%A Arsenii A. Sagdeev
%A Andrei M. Raigorodskii
%A Arsenii A. Sagdeev
%A Andrei M. Raigorodskii
%T On a Frankl-Wilson theorem and its geometric corollaries
%J Acta mathematica Universitatis Comenianae
%D 2019
%P 1029-1033
%V 88
%N 3
%U http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a104/
%F AMUC_2019_88_3_a104
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 $(\xi_m+o(1))^n$.
