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Finite embeddability of sets and ultrafilters. (English) Zbl 1382.03068

Summary: 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.

MSC:

03E05 Other combinatorial set theory
03H15 Nonstandard models of arithmetic
03C20 Ultraproducts and related constructions
54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.)
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