Unavoidable subhypergraphs: a-clusters

Zoltán Füredi1 Alfréd Rényi Institute Hungarian Academy of Sciences Budapest, P.O.Box 127, Hungary, H-1364, [email protected] and Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, USA [email protected] Lale Özkahya Department of Mathematics Iowa State University Ames, IA 50011, USA [email protected] Abstract 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 su ciently large n, if F is a k-uniform family on n vertices with | F | exceeding the Erd®s-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.