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 is an element of S, we have vertical bar i - j vertical bar >= s. We define almost s-stable Kneser hypergraph KG(r)([n]k)(s-stab)(similar to) 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)(s-stab)(similar to) is equal to the chromatic number of the usual Kneser hypergraphs KG(p)([n]k), namely that it is equal to inverted right perpendicular n-(k-1)p/p-1 inverted left perpendicular. 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. (C) 2011 Elsevier Inc. All rights reserved.