Hypergraphs with finitely many isomorphism subtypes

Abstract:

Let $\mathcal {H} = (H,E)$ be an $n$-uniform infinite hypergraph such that the number of isomorphism types of induced subgraphs of $\mathcal {H}$ of cardinality $\lambda$ is finite for some infinite $\lambda$. 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.