Unavoidable subhypergraphs: a-clusters

Abstract One of the central problems of extremal hypergraph theory is the description of unavoidable subhypergraphs, in other words, the Turan problem. Let a = ( a 1 , … , a p ) be a sequence of positive integers, p ⩾ 2 , k = a 1 + … + a p . An a-cluster is a family of k-sets { F 0 , … , F p } such that the sets F i \ F 0 are pairwise disjoint ( 1 ⩽ i ⩽ p ) , | F i \ F 0 | = a i , and the sets F 0 \ F i are pairwise disjoint, too. Given a there is a unique a-cluster, and the sets F 0 \ F i form an a-partition of F 0 . With an intensive use of the delta-system method we prove that for k > p > 1 and sufficiently large n, ( n > n 0 ( k ) ) , if F is an n-vertex k-uniform family 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.