Unavoidable subhypergraphs: a-clusters

One of the central problems of extremal hypergraph theory is the description of unavoidable subhypergraphs, in other words, the Turán problem. Let a = ( a 1 , … , a p ) be a sequence of positive integers, k = a 1 + ⋯ + a p . An a-partition of a k-set F is a partition in the form F = A 1 ∪ ⋯ ∪ A p with | A i | = a i for 1 ⩽ i ⩽ p . An a-cluster A with host F 0 is a family of k-sets { F 0 , … , F p } such that for some a-partition of F 0 , F 0 ∩ F i = F 0 ∖ A i for 1 ⩽ i ⩽ p and the sets F i ∖ F 0 are pairwise disjoint. The family A has 2k vertices and it is unique up to isomorphisms. With an intensive use of the delta-system method we prove that for k > p and sufficiently large n, if F is a k-uniform family on n vertices with | F | exceeding the Erdős–Ko–Rado bound ( n − 1 k − 1 ) , then F contains an a-cluster. The only extremal family consists of all the k-subsets containing a given element.