On the chromatic number of generalized Kneser hypergraphs

Abstract

The generalized Kneser hypergraph KGr(n,k,s) is the hypergraph whose vertices are all the k-subsets of {1,…,n}, and edges are r-tuples of distinct vertices such that any pair of them has at most s elements in their intersection. In this note, we show that for each non-negative integers k,n,r,s satisfying n≥r(k−1)+1, k>s≥0, and r≥2, we have χ(KGr(n,k,s))≥n−r(k−s−1)r−1,which extends the previously known result by Alon–Frankl–Lovász.