Embedding dualities for set partitions and for relational structures

We show that for a set FF of forbidden set partitions and an integer kk there is a finite collection DD of partitions of ordinals, such that any finite partition with at most kk blocks avoids all the elements of FF if and only if it is contained in at least one element of DD. Using this result, we reprove rationality of the generating function enumerating a hereditary class of set partitions with a bounded number of blocks. We show that this result does not extend to partitions with an unbounded number of blocks.We also consider hereditary classes of relational structures. We give a characterization of those classes that can be expressed as classes of finite substructures of a finite collection of (possibly infinite) relational structures.