Hypergraph Turán numbers of linear cycles

摘要

A k-uniform linear cycle of length ?,denoted by C_?~(k), is a cyclic list of k-sets A_1... A_? such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k ≥ 5 and ?≥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 ?. For odd ?=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 ? = 2t + 2, the unique extremal family consists of F_s 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 Erd?s's result on largest k-uniform families without t + 1 pairwise disjoint members and confirm, in a stronger form, a conjecture of Mubayi and Verstra?te. Our main method is the delta system method.