On 3-uniform hypergraphs without a cycle of a given length

We study the maximum number of hyperedges in a 3-uniform hypergraph on n vertices that does not contain a Berge cycle of a given length ℓ. In particular we prove that the upper bound for C2k+1-free hypergraphs is of the order O(k2n1+1/k), improving the upper bound of Győri and Lemons (2012) by a factor of Θ(k2). Similar bounds are shown for linear hypergraphs.