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=(a1,. .,ap) be a sequence of positive integers, k=a1+...+ap. An a- partition of a k-set F is a partition in the form F=A1∪...∪Ap with |Ai|=ai for 1≤i≤p. An a- cluster A with host F0 is a family of k-sets {F0,. .,Fp} such that for some a-partition of F0, F0∩Fi=F0≙Ai for 1≤i≤p and the sets Fi\F0 are pairwise disjoint. The family A has 2. k 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 Erdos-Ko-Rado bound (n-1k-1), then F contains an a-cluster. The only extremal family consists of all the k-subsets containing a given element.