$3$-uniform hypergraphs without a cycle of length five

In this paper we show that the maximum number of hyperedges in a $3$-uniform hypergraph on $n$ vertices without a (Berge) cycle of length five is less than $(0.254+o(1))n^{3\mathrm{/}2}$, improving an estimate of Bollob'as and Gy\H{o}ri. We obtain this result by showing that not many $3$-paths can start from certain subgraphs of the shadow.