A counterexample to a conjecture on the chromatic number of r $r$‐stable Kneser hypergraphs

Abstract

AbstractLet be a set of subsets of and be an integer. The ‐colorability defect of is the least number of elements of that need to be removed such that the remaining ground set can be ‐colored without inducing monochromatic sets in . The set is a subset of containing all elements of such that any two members of are at least at distance apart on the ‐cycle. The main purpose of this note is to give a counterexample to a conjecture raised by Florian Frick, which says if is a set of subsets of and , then the number of blocks needed to partition the elements of such that no pairwise disjoint sets lie in the same block is at least .