A k-path is a hypergraph such that if and otherwise. A k-cycle is a hypergraph obtained from a -path by adding an edge that shares one vertex with , another vertex with and is disjoint from the other edges.
Let 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 , and , for large enough n: and we characterize all the extremal r-graphs. We also solve the case , which needs a special treatment. The case 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 1970s. In particular, we extend the work (and settle a conjecture) of Füredi, Jiang and Seiver who solved this problem for when and of Füredi and Jiang who solved it for when . They used the delta system method, while we use a novel approach which involves random sampling from the shadow of an r-graph.