Hypergraphs with finitely many isomorphism subtypes

Let K = (H, E) be an n-uniform infinite hypergraph such that the number of isomorphism types of induced subgraphs of K of cardinality λ is finite for some infinite λ. We solve a problem due independently to Jamison and Pouzet, by showing that there is a finite subset K of H such that the induced subgraph on H - K is either empty or complete. We also characterize such hypergraphs in terms of finite (not necessarily uniform) hypergraphs.