A d-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 Chvatal and a recent conjecture of Mubayi. For d > 1 and c>0 there is a number T such that the following holds for n large and cn < k < n/2 - T. The unique largest k-uniform set system on [n]={1,...,n} that does not contain a d-simplex consists of all k-sets containing a specific element. The same holds when "d-simplex" is replaced by "d-cluster".
In the non-uniform setting we obtain the following result that generalises a question of Erdos and a result of Milner, who proved the case d=2. Suppose d > 1 and n is large. Then the unique largest set system on [n] that does not contain a d-dimensional simplex consists 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 result that we obtain for intersecting families, which supersedes a result of Friedgut that was proved using spectral techniques, and is based on a purely combinatorial result of Frankl.
This is joint work with Dhruv Mubayi.