  The celebrated Erd\H{o}s-Hajnal conjecture says that any graph without a
fixed induced subgraph $H$ contains a very large homogeneous set. A direct
analog of this conjecture is not true for hypergraphs. In this paper we present
two natural variants of this problem which do hold for hypergraphs. We show
that for every $r \geq 3$, $m \geq m_0(r)$ and $0 \leq f \leq \binom{m}{r}$, if
an $r$-graph $G$ does not contain $m$ vertices spanning exactly $f$ edges, then
$G$ contains much bigger homogeneous sets than what is guaranteed to exist in
general $r$-graphs. We also prove that if a $3$-graph $G$ does not contain
homogeneous sets of polynomial size, then for every $m \geq 3$ there are
$\Omega(m^3)$ values of $f$ such that $G$ contains $m$ vertices spanning
exactly $f$ edges. This makes progress on a conjecture of Axenovich,
Brada\v{c}, Gishboliner, Mubayi and Weber.
