Turán problems and shadows I: Paths and cycles

A k-path is a hypergraph Pk = {e1, e2, . . . , ek} such that |ei ∩ ej | = 1 if |j − i| = 1 and ei ∩ ej = ∅ otherwise. A k-cycle is a hypergraph Ck = {e1, e2, . . . , ek} obtained from a (k − 1)path {e1, e2, . . . , ek−1} by adding an edge ek that shares one vertex with e1, another vertex with ek−1 and is disjoint from the other edges. Let exr(n,G) be the maximum number of edges in an r-graph with n vertices not containing a given r-graph G. We prove that for fixed r ≥ 3, k ≥ 4 and (k, r) 6= (4, 3), for large enough n: exr(n, Pk) = exr(n,Ck) = ( n r ) − ( n− bk−1 2 c r ) + { 0 if k is odd ( n−b k−1 2 c−2 r−2 ) if k is even and we characterize all the extremal r-graphs. We also solve the case (k, r) = (4, 3), which needs a special treatment. The case k = 3 was settled by Frankl and Füredi. This work is the next step in a long line of research beginning with conjectures of Erdős and Sós from the early 1970’s. In particular, we extend the work (and settle a conjecture) of Füredi, Jiang and Seiver who solved this problem for Pk when r ≥ 4 and of Füredi and Jiang who solved it for Ck when r ≥ 5. They used the delta system method, while we use a novel approach which involves random sampling from the shadow of an r-graph.