Hypergraphs with finitely many isomorphism subtypes

Abstract

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