Hypergraph Turan numbers of linear cycles

A k-uniform linear cycle of length P, denoted by etk), is a cyclic list of k-sets A1, . . . , Al such that consecutive sets intersect fn exactly one element and nonconsecutive sets are disjoint. For all k <= 5 and l >= 3 and sufficiently large n we determine the largest size of a k-uniform set family on [n] not containing a linear cycle of length P. For odd e = 2t 1 the unique extrema' family F-S consists of all k-sets in [n] intersecting a fixed t-set S in [n]. For even P = 2t + 2, the unique extremal family consists of Ts plus all the k-sets outside S containing some fixed two elements. For k >= 4 and large n we also establish an exact result for so-called minimal cycles. For all k >= 4 our results substantially extend Erdos's result on largest k-uniform families without t + 1 pairwise disjoint members and confirm, in a stronger form, a conjecture of Mubayi and Verstraete. Our main method is the delta system method. (C) 2014 Elsevier.Inc. All rights reserved.