The chromatic number of almost stable Kneser hypergraphs

Let V (n, k, s) be the set of k-subsets S of [n] such that for all i, j ∈ S, we have |i−j| ≥ s We define almost s-stable Kneser hypergraph KG ( [n] k )∼ s-stab to be the r-uniform hypergraph whose vertex set is V (n, k, s) and whose edges are the r-uples of disjoint elements of V (n, k, s). With the help of a Zp-Tucker lemma, we prove that, for p prime and for any n ≥ kp, the chromatic number of almost 2-stable Kneser hypergraphs KG ( [n] k )∼ 2-stab is equal to the chromatic number of the usual Kneser hypergraphs KG ( [n] k ) , namely that it is equal to ⌈ n−(k−1)p p−1 ⌉ . Related results are also proved, in particular, a short combinatorial proof of Schrijver’s theorem (about the chromatic number of stable Kneser graphs) and some evidences are given for a new conjecture concerning the chromatic number of usual s-stable r-uniform Kneser hypergraphs.