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 $\ell$. In particular we prove that the upper bound for $C_{2k+1}$-free hypergraphs is of the order $O(k^2n^{1+1/k})$, improving the upper bound of Gy\"ori and Lemons by a factor of $\Theta(k^2)$. Similar bounds are shown for linear hypergraphs.