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Let $r \\ge 3$, $k \\ge 3$, and suppose that $n \\ge (r-2)(k-2)+1$. If $H$ is a linear $r$-uniform hypergraph on $n$ vertices and \\[|E(H)| \\geq \\frac{k-2}{r^2((r-2)(k-2)+1)}n^2 + \\frac{n}{r},\\] then $H$ contains $k$ edges spanning at most $(r-2)k+3$ vertices. In the standard linear-density normalization, this gives the asymptotic density threshold $c \\geq \\frac{r-1}{r} \\cdot \\frac{k-2}{(r-2)(k-2)+1} + o(1)$. 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