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 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 , then F contains an a-cluster. The only extremal family consists of all the k-subsets containing a given element.