Hypergraph Turلn numbers of linear cycles

A k-uniform linear cycle of length ℓ, denoted by C ℓ ( k ) , is a cyclic list of k-sets A 1 , … , A ℓ such that consecutive sets intersect in 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 ℓ. For odd ℓ = 2 t + 1 the unique extremal family F S consists of all k-sets in [ n ] intersecting a fixed t-set S in [ n ] . For even ℓ = 2 t + 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 Erdősʼs result on largest k-uniform families without t + 1 pairwise disjoint members and confirm, in a stronger form, a conjecture of Mubayi and Verstraëte. Our main method is the delta system method.