New upper bound for the chromatic number of a random subgraph of a distance graph

Abstract

This paper is related to the classical Hadwiger-Nelson problem dealing with the chromatic numbers of distance graphs in ℝn. We consider the class consisting of the graphs G(n, 2s + 1, s) = (V (n, 2s + 1), E(n, 2s + 1, s)) defined as follows: $$V(n,2s + 1) = \{ x = (x_1 ,x_2 ,...,x_n ):x_i \in \{ 0,1\} ,x_1 + x_2 + \cdots + x_n = 2s + 1\} ,E(n,2s + 1,s) = \{ \{ x,y\} :(x,y) = s\} ,$$ where (x, y) stands for the inner product. We study the random graph G(G(n, 2s + 1, s), p) each of whose edges is taken from the set E(n, 2s+1, s) with probability p independently of the other edges. We prove a new bound for the chromatic number of such a graph.