Turán problems and shadows I: Paths and cycles

A k-path is a hypergraph P_k={e₁, e₂,..., e_k} such that |e_i∩e_j|=1 if |j-i|=1 and e_i∩e_j=θ otherwise. A k-cycle is a hypergraph C_k={e₁, e₂,..., e_k} obtained from a (k-1)-path {e₁, e₂,..., e_k-1} by adding an edge e_k that shares one vertex with e₁, another vertex with e_k-1 and is disjoint from the other edges. Let ex_r(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)≠(4, 3), for large enough n: (equations) if k is odd, 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.