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 to be the r-uniform hypergraph whose vertex set is V(n,k,s) and whose edges are the r-tuples 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 is equal to the chromatic number of the usual Kneser hypergraphs , namely that it is equal to .Related results are also proved, in particular, a short combinatorial proof of Schrijverʼs 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.