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|εs. We define almost s-stable Kneser hypergraph KG^r(^[n]_k)s-stab_̃ 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 Z_p-Tucker lemma, we prove that, for p prime and for any n≥kp, the chromatic number of almost 2-stable Kneser hypergraphs KG^p(^[n]_k)2-stab^̃ is equal to the chromatic number of the usual Kneser hypergraphs KG^p(^[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.