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] kwith empty intersection and union of size no more than 2k. Mubayi [6] conjectured that the largest size of a d-cluster-free family F ⊂ [n] k is n−1 k−1 , with equality holding only for a maximum-sized star. Here we prove two results. The first resolves Mubayi's conjecture and proves a stronger result, thus completing a new generalization of the Erdős-Ko-Rado Theorem. The second shows, by a different technique, that for a slightly more limited set of parameters only a very specific kind of d-cluster need be forbidden to achieve the same bound.

New results on simplex-clusters in set systems

“…As with simplices, it was conjectured [13] that a family F ⊆ [n] k containing no d-cluster would have to obey the bound |F | ≤ n−1 k−1 . This problem also had a long history (see [6,13,14,15,10] for some of the more significant developments) and was completely resolved recently in a paper of the author [3]. In 2010, Keevash and Mubayi extended both conjectures by hypothesizing that the same bound would hold for any F ⊆ [n] k containing no d-simplex-cluster, and very recently Lifshitz answered their question in the affirmative for all n > n 0 (d) in [12].…”

“…In our proof of Theorem 2, we will use as one of our primary tools the following Theorem of the author, which was used in [3] to resolve the question of d-clusters. We include the proof for completeness:…”