The well-known Erdős-Hajnal conjecture states that for any graph F, there exists ϵ>0 such that every n-vertex graph G that contains no induced copy of F has a homogeneous set of size at least nϵ. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid 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 edges for any positive m and 0≤f≤(m2), then we obtain large homogeneous sets. For triple systems, in the first nontrivial case m=4, for every S⊆0,1,2,3,4, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in S. In most cases the bounds are essentially tight. We also determine, for all S, whether the growth rate is polynomial or polylogarithmic. Some open problems remain