Set systems without a simplex or a cluster

Abstract

A d-dimensional simplex is a collection of d+1 sets with empty intersection, every d of which have nonempty intersection. A k-uniform d-cluster is a collection of d+1 sets of size k with empty intersection and union of size at most 2k.

We prove the following result which simultaneously addresses an old conjecture of Chvátal [6] and a recent conjecture of the second author [28]. For d≥2 and ζ >0 there is a number T such that the following holds for sufficiently large n. Let G be a k-uniform set system on [n] ={1,…,n} with ζ n<k <n/2−T, and suppose either that G contains no d-dimensional simplex or that G contains no d-cluster. Then |G|≤$\left( {\begin{array}{*{20}c} {n - 1} \\ {k - 1} \\ \end{array} } \right)$ with equality only for the family of all k-sets containing a specific element.

In the non-uniform setting we obtain the following exact result that generalises a question of Erdős and a result of Milner, who proved the case d=2. Suppose d≥2 and G is a set system on [n] that does not contain a d-dimensional simplex, with n sufficiently large. Then |G|≤2ⁿ⁻¹+Σ ^d−1_i=0 $\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right)$, with equality only for the family consisting of all sets that either contain some specific element or have size at most d−1.

Each of these results is proved via the corresponding stability result, which gives structural information on any G whose size is close to maximum. These in turn rely on a stability theorem that we obtain using an earlier result of Frankl.