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We say that $\\mathcal{H}$ contains a Berge-$F$ if there exist injections $\\psi:U\\to V$ and $\\varphi:E\\to \\mathcal{E}$ such that for every $e=\\{u,v\\}\\in E$, $\\{\\psi(u),\\psi(v)\\}\\subset\\varphi(e)$. Let $ex_r(n,F)$ denote the maximum number of hyperedges in an $r$-uniform hypergraph on $n$ vertices which does not contain a Berge-$F$.\n  For small enough $r$ and non-bipartite $F$, $ex_r(n,F)=\\Omega(n^2)$; we show that for sufficiently large $r$, $ex_r(n,F)=o(n^2)$. 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