Relational structures having finitely many full-cardinality restrictions

Considering an arbitrary relational structure on an infinite groundset, we analyze the implications of the following finiteness hypothesis (H): for some infinite cardinality μ there exist only finitely many isomorphism types of substructures of size μ. We show that the class C of relational structures satisfying (H) is intimately related to an explicit family of linear orders. Based on this, we show how to construct every member of C, up to isomorphism, thereby describing C completely. As an application, we characterize the profile of a relational structure that satisfies (H). Our work extends earlier published results concerning the special case of hypergraphs.