Subgraphs with large minimum ℓ-degree in hypergraphs where almost all ℓ-degrees are large

Let G be an r-uniform hypergraph on n vertices such that all but at most $\varepsilon \binom{n}{\ell}$ $\ell$-subsets of vertices have degree at least $p \binom{n-\ell}{r-\ell}$. We show that G contains a large subgraph with high minimum ℓ-degree.