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The weight of a non-uniform hypergraph $\\mathcal{H}$ is the quantity $\\sum_{h \\in E(\\mathcal{H})} |h|$.\n  Suppose $\\mathcal{H}$ is a Berge-$F$-free hypergraph on $n$ vertices. In this short note, we prove that as long as every edge of $\\mathcal{H}$ has size at least the Ramsey number of $F$ and at most $o(n)$, the weight of $\\mathcal{H}$ is $o(n^2)$. 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