On the Chromatic Number of Generalized Stable Kneser Graphs

For each integer triple (n, k, s) such that k ≥ 2, s ≥ 2, and n ≥ ks, define a graph in the following manner. The vertex set consists of all k-subsets S of Zn such that any two elements in S are on circular distance at least s. Two vertices form an edge if and only if they are disjoint. For the special case s = 2, we get Schrijver’s stable Kneser graph. The general construction is due to Meunier, who conjectured that the chromatic number of the graph is n − s(k − 1). By a famous result due to Schrijver, the conjecture is known to be true for s = 2. The main result of the present paper is that the conjecture is true for s ≥ 4, provided n is sufficiently large in terms of s and k. The proof techniques do not apply to the case s = 3, which remains nearly completely open.