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