The chromatic number of almost stable Kneser hypergraphs

Abstract

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 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 ⩾ k p , 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 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.