  The Erd\H{o}s--Hajnal Theorem asserts that non-universal graphs, that is,
graphs that do not contain an induced copy of some fixed graph $H$, have
homogeneous sets of size significantly larger than one can generally expect to
find in a graph. We obtain two results of this flavor in the setting of
$r$-uniform hypergraphs.
  A theorem of R\"odl asserts that if an $n$-vertex graph is non-universal then
it contains an almost homogeneous set (i.e one with edge density either very
close to $0$ or $1$) of size $\Omega(n)$. We prove that if a $3$-uniform
hypergraph is non-universal then it contains an almost homogeneous set of size
$\Omega(\log n)$. An example of R\"odl from 1986 shows that this bound is
tight.
  Let $R_r(t)$ denote the size of the largest non-universal $r$-graph $G$ so
that neither $G$ nor its complement contain a complete $r$-partite subgraph
with parts of size $t$. We prove an Erd\H{o}s--Hajnal-type stepping-up lemma,
showing how to transform a lower bound for $R_{r}(t)$ into a lower bound for
$R_{r+1}(t)$. As an application of this lemma, we improve a bound of
Conlon--Fox--Sudakov by showing that $R_3(t) \geq t^{\Omega(t)}$.

Combinatorics