Hypergraph Turan numbers of linear cycles

Abstract

A k-uniform linear cycle of length s is a cyclic list of k-sets A_1,..., A_s such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k at least 5 and s at least 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 s. For odd s=2t+1 the unique extremal family F_S consists of all k-sets in [n] intersecting a fixed t-set S in [n]. For even s=2t+2, the unique extremal family consists of F_S plus all the k-sets outside S containing some fixed two elements. For k at least 4 and large n we also establish an exact result for so-called minimal cycles. For all k at least 4 our results substantially extend Erdos' 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.