  Given a family of hypergraphs $\mathcal H$, let $f(m,\mathcal H)$ denote the
largest size of an $\mathcal H$-free subgraph that one is guaranteed to find in
every hypergraph with $m$ edges. This function was first introduced by
Erd\H{o}s and Koml\'{o}s in 1969 in the context of union-free families, and
various other special cases have been extensively studied since then. In an
attempt to develop a general theory for these questions, we consider the
following basic issue: which sequences of hypergraph families $\{\mathcal
H_m\}$ have bounded $f(m,\mathcal H_m)$ as $m\to\infty$? A variety of bounds
for $f(m,\mathcal H_m)$ are obtained which answer this question in some cases.
Obtaining a complete description of sequences $\{\mathcal H_m\}$ for which
$f(m,\mathcal H_m)$ is bounded seems hopeless.
