New Results on Simplex-Clusters in Set Systems

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 d -cluster. The Erdös-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 /n -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 and n ≥ 2k - d + 2, which in turn resolves all remaining cases of the Erdös-Chvatal Conjecture except when n is very small (i.e. n < 2k-d + 2).