On r-uniform hypergraphs with circumference less than r

We show that for each k≥4 and n>r≥k+1, every n-vertex hypergraph with edge sizes at least r and no Berge cycle of length at least k has at most [Formula presented] edges. The bound is exact, and we describe the extremal hypergraphs. This implies and slightly refines the theorem of Győri, Katona and Lemons that for n>r≥k≥3, every n-vertex r-uniform hypergraph with no Berge path of length k has at most [Formula presented] edges. To obtain the bounds, we study bipartite graphs with no cycles of length at least 2k, and then translate the results into the language of multi-hypergraphs.