Hypergraph supersaturation by sampling ($e(G)\ge (\pi (H)+ϵ)\left(\genfrac{}{}{0ex}{}{n}{\ell }\right)⟹\#H\ge \delta n^{|H|}$)

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.