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The well-known Erd\H{o}s-Hajnal conjecture states that for any graph $F$,there exists $\epsilon>0$ such that every $n$-vertex graph $G$ that contains noinduced copy of $F$ has a homogeneous set of size at least $n^{\epsilon}$. Weconsider a variant of the Erd\H{o}s-Hajnal problem for hypergraphs where weforbid a family of hypergraphs described by their orders and sizes. For graphs,we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edgesfor any positive $m$ and $0\leq f \leq \binom{m}{2}$, then we obtain largehomogeneous sets. For triple systems, in the first nontrivial case $m=4$, forevery $S \subseteq \{0,1,2,3,4\}$, we give bounds on the minimum size of ahomogeneous set in a triple system where the number of edges spanned by everyfour vertices is not in $S$. In most cases the bounds are essentially tight. Wealso determine, for all $S$, whether the growth rate is polynomial orpolylogarithmic. Some open problems remain.

Large cliques or co-cliques in hypergraphs with forbidden order-size pairs