Finite Embeddability of Sets and Ultrafilters

A set $A$ of natural numbers is finitely embeddable in another such set $B$ if every finite subset of $A$ has a rightward translate that is a subset of $B$. This notion of finite embeddability arose in combinatorial number theory, but in this paper we study it in its own right. We also study a related notion of finite embeddability of ultrafilters on the natural numbers. Among other results, we obtain connections between finite embeddability and the algebraic and topological structure of the Stone–Čech compactification of the discrete space of natural numbers. We also obtain connections with nonstandard models of arithmetic.