On the chromatic numbers of small-dimensional Euclidean spaces

TL;DR

This paper investigates the chromatic numbers of small-dimensional Euclidean spaces by analyzing specific graph sequences, determining independence numbers, and deriving new lower bounds for chromatic numbers in these spaces.

Contribution

It precisely computes the independence number of a particular graph sequence and establishes new lower bounds for the chromatic numbers of Euclidean and rational spaces.

Findings

Exact independence number of G_n determined

New lower bounds for chi(R^n) established

Lower bounds for chi(Q^n) derived

Abstract

The paper is devoted to the study of graph sequence G_n = (V_n, E_n) where V_n is the set of all vectors v in R^n with coordinates from {-1, 0, 1} such that |v| = sqrt(3), and E_n consists of all pairs of vertices with the scalar product 1. We find exactly the independence number of G_n. As a corollary we get some new lower bounds of chi(\R^n) and chi(\Q^n) for small values of n.