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 Györi and Lemons by a factor of $Θ(k^2)$. Similar bounds are shown for linear hypergraphs.

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