Blass, Andreas; Di Nasso, Mauro Finite embeddability of sets and ultrafilters. (English) Zbl 1382.03068 Bull. Pol. Acad. Sci., Math. 63, No. 3, 195-206 (2015). 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. Cited in 1 ReviewCited in 5 Documents 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.) Keywords:ultrafilter; nonstandard models; shift map PDFBibTeX XMLCite \textit{A. Blass} and \textit{M. Di Nasso}, Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068) Full Text: DOI arXiv References: [1] [1]M. Beiglb\"{}ock, An ultrafilter approach to Jin’s theorem, Israel J. Math. 185 (2011), 369–374. · Zbl 1300.11015 [2] [2]C. C. Chang and H. J. Keisler, Model Theory, 3rd ed., North-Holland, 1990. [3] [3]M. Davis, Applied Nonstandard Analysis, Wiley, 1977. [4] [4]M. Di Nasso, Embeddability properties of difference sets, Integers 14 (2014), no. A27. [5] [5]N. Hindman and D. Strauss, Algebra in the Stone– \check{}Cech Compactification. Theory and Applications, 2nd ed., de Gruyter, 2012. · Zbl 1241.22001 [6] [6]P. Krautzberger, Idempotent filters and ultrafilters, Ph.D. thesis, Freie Univ. Berlin, 2009. · Zbl 1354.03062 [7] [7]L. Luperi Baglini, Hyperintegers and nonstandard techniques in combinatorics of numbers, Ph.D. thesis, Univ. di Siena, 2012. [8] [8]L. Luperi Baglini, Ultrafilters maximal for finite embeddability, J. Logic Anal. 6 (2014), no. 6, 1–16. · Zbl 1321.05011 [9] [9]I. Z. Ruzsa, On difference sets, Studia Sci. Math. Hungar. 13 (1978), 319–326. · Zbl 0423.10027 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.