On the d-cluster generalization of Erdős-Ko-Rado

Abstract

If 2≤d≤k and n≥dk/(d−1), a d-cluster is defined to be a collection of d elements of ([n]k) with empty intersection and union of size no more than 2k. Mubayi [14] conjectured that the largest size of a d-cluster-free family F⊂([n]k) is (n−1k−1), with equality holding only for a maximum-sized star. Here, we resolve Mubayi's conjecture and prove a slightly stronger result, thus completing a new generalization of the Erdős-Ko-Rado Theorem.