Lower bounds for the chromatic number of certain Kneser-type hypergraphs

Abstract

Let n≥1, r≥2, and s≥0 be integers and P={P1,…,Pl} be a partition of [n]={1,…,n} with |Pi|≤r for i=1,…,l. Also, let F be a family of non-empty subsets of [n]. The r-uniform Kneser-type hypergraph KGr(F,P,s) is the hypergraph with the vertex set consisting of all P-admissible elements F∈F, that is |F∩Pi|≤1 for i=1,…,l and the edge set consisting of all r-subsets {F1,…,Fr} of the vertex set that |Fi∩Fj|≤s for all 1≤i<j≤r. In this article, we extend the equitable r-colorability defect ecdr(F) of Abyazi Sani and Alishahi to the case when one allows intersections among the vertices of an edge. It will be denoted by ecdr(F,s). We then prove that the chromatic number of KGr(F,P,s) is bounded from below by ecdr(F,⌊s/2⌋)r−1, under the technical assumption that the pair (F,P) is ⌊s/2⌋-good. This condition holds in the cases when s=0, or P consists of singletons, that is P-admissibility is automatically satisfied for all sets or the family F is the family of all k-sets in [n] for some integer s<k<n. This work generalizes many existing results in the literature on the Kneser hypergraphs. It generalizes the previous results of the current authors from the special family of all k-subsets of [n] to a general family F of subsets.