On the chromatic numbers of small-dimensional Euclidean spaces

This paper is devoted to the study of the graph sequence Gn=(Vn,En), where Vn is the set of all vectors v∈Rn with coordinates in {−1,0,1} such that |v|=3 and En consists of all pairs of vertices with scalar product 1. We find the exact value of the independence number of Gn. As a corollary we get new lower bounds on χ(Rn) and χ(Qn) for small values of n.