= Hypergraph supersaturation by sampling
{title2=$e(G)\ge(\pi(H)+\epsilon)\binom n\ell\Longrightarrow\#H\ge\delta n^{|H|}$}
Normalized hypergraph extremal numbers decrease under <vertex> sampling and converge to <Turán density>. At a fixed sample size where the extremal density is close to that limit, excess density forces a positive fraction of samples to contain $H$. Counting incidences of copies and samples proves the displayed lower bound. Shrinking $\delta$ also handles the finitely many small orders when the required count is rounded down.
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