New results on simplex-clusters in set systems

Abstract

A $d$-simplex is defined to be a collection $A_1,\dots,A_{d+1}$ 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. Furthermore, a $d$-cluster is a collection of $d+1$ such sets with empty intersection and union of size $\le 2k$, and a $d$-simplex-cluster is such a collection that is both a $d$-simplex and a $d$-cluster. The Erd\H{o}s-Chv\'{a}tal $d$-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 \choose 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 $4 \le d+1 \le k$ and $n \ge 2k-d+2$, which in turn resolves all remaining cases of the Erd\H{o}s-Chv\'{a}tal Conjecture except when $n$ is very small (i.e. $n < 2k-d+2$).