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 2 k + 1 has less than 4 k 4 n 1 + 1 / k + O ( n ) edges. Constructions show that these bounds are best possible (up to constant factor) for k = 1 , 2 , 3 , 5 .