New results on simplex-clusters in set systems

Abstract

A d-simplex is defined to be a collection A 1 , . . . , A d+1 of subsets of size k of [n] such that the intersection of all of them is empty, but the intersection of any d of them is non-empty. Furthemore, a d-cluster is a collection of d + 1 such sets with empty intersection and union of size ≤ 2k, and a d-simplex-cluster is such a collection that is both a d-simplex and a dcluster. The Erdős-Chvátal d-simplex Conjecture from 1974 states that any family of k-subsets of [n] containing no d-simplex must be of size no greater than n−1 k−1 . In 2011, Keevash and Mubayi extended this conjecture by hypothesizing that the same bound would hold for families containing no d-simplex-cluster. In this paper, we resolve Keevash and Mubayi's conjecture for all 4 ≤ d + 1 ≤ k ≤ n/2, which in turn resolves all remaining cases of the Erdős-Chvátal Conjecture except when n is very small (i.e. n < 2k).

A note on hypergraphs without non-trivial intersecting subgraphs

“…Mubayi and Verstraëte [17] proved the conjecture for all k ≥ 3 and d = 2. Recently, Currier [4] proved this conjecture for all k ≥ d + 1 ≥ 3 and n ≥ 2k. The Chvátal Simplex Conjecture is still open in general for n < 2k and 3 ≤ d ≤ k − 2, and we refer the reader to [1,3,8,5,6,12,13,15] and their references for more results related to this conjecture.…”