Hypergraphs with Few Berge Paths of Fixed Length between Vertices

In this paper we study the maximum number of hyperedges which may be in an r-uniform hypergraph under the restriction that no pair of vertices has more than t Berge paths of length k between them. When r = t = 2, this is the even-cycle problem asking for ex(n, C-2k). We extend results of Fiiredi and Simonovits and of Conlon, who studied the problem when r = 2. In particular, we show that for fixed k and r, there is a constant t such that the maximum number of edges can be determined in order of magnitude.