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The linear Tur\\'an number of $\\mathcal F$ is the maximum possible number of edges in a $3$-uniform linear hypergraph on $n$ vertices which contains no member of $\\mathcal{F}$ as a subhypergraph.\n  In this paper we show that the linear Tur\\'an number of the five cycle $C_5$ (in the Berge sense) is $\\frac{1}{3 \\sqrt3}n^{3/2}$ asymptotically. 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