On set systems without a simplex-cluster and the junta method

A family {A₀,…,A_d} of k-element subsets of [n]={1,2,…,n} is called a d-simplex-cluster if A₀∩⋯∩A_d=∅, |A₀∪⋯∪A_d|≤2k, and the intersection of any d of the sets in {A₀,…,A_d} is nonempty. In 2010, Keevash and Mubayi conjectured that for any [Formula presented], the largest family of k-element subsets of [n] that does not contain a d-simplex-cluster is the family of all k-subsets that contain a given element. We prove the conjecture for all k≥ζn for an arbitrarily small ζ>0, provided that n≥n₀(ζ,d). We call a family {A₀,…,A_d} of k-element subsets of [n] a (d,k,s)-cluster if A₀∩⋯∩A_d=∅ and |A₀∪⋯∪A_d|≤s. We also show that for any [Formula presented] the largest family of k-element subsets of [n] that does not contain a [Formula presented]-cluster is again the family of all k-subsets that contain a given element, provided that n≥n₀(ζ,d). Our proof is based on the junta method for extremal combinatorics initiated by Dinur and Friedgut and further developed by Ellis, Keller, and the author.