Hypergraphs with no odd cycle of given length

We give upper bounds for the size of 3-uniform hypergraphs avoiding a given odd cycle using the definition of a cycle due to Berge. In particular, we show that a 3-uniform hypergraph containing no cycle of length 2k+1 has less than 4k4n1+1/k+O(n) edges. Constructions show that these bounds are best possible (up to constant factor) for k=1,2,3,5.