Unavoidable subhypergraphs: a-clusters

One of the central problems of extremal hypergraph theory is the description of unavoidable subhypergraphs, in other words, the Turan problem. Let a=(a1,…,ap) be a sequence of positive integers, p⩾2, k=a1+…+ap. An a-cluster is a family of k-sets {F0,…,Fp} such that the sets Fi\F0 are pairwise disjoint (1⩽i⩽p), |Fi\F0|=ai, and the sets F0\Fi are pairwise disjoint, too. Given a there is a unique a-cluster, and the sets F0\Fi form an a-partition of F0. With an intensive use of the delta-system method we prove that for k>p>1 and sufficiently large n, (n>n0(k)), if F is an n-vertex k-uniform family with |F| exceeding the Erdős-Ko-Rado bound , then F contains an a-cluster. The only extremal family consists of all the k-subsets containing a given element.