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A dichotomy theorem for subsets of the power set of the natural numbers


Author: I. Gasparis
Journal: Proc. Amer. Math. Soc. 129 (2001), 759-764
MSC (1991): Primary 46B03; Secondary 06A07, 03E02
DOI: https://doi.org/10.1090/S0002-9939-00-05594-5
Published electronically: August 30, 2000
MathSciNet review: 1707146
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Abstract:

The following dichotomy is established for any pair \(\mathcal{F}\), \(\mathcal{G}\) of hereditary families of finite subsets of \(\mathbb{N}\): Given \(N\), an infinite subset of \(\mathbb{N}\), there exists \(M\) an infinite subset of \(N\) so that either \(\mathcal{G} \cap [M]^{< \infty} \subset \mathcal{F}\), or \(\mathcal{F} \cap [M]^{< \infty} \subset \mathcal{G}\), where \([M]^{< \infty}\) denotes the set of all finite subsets of \(M\).


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Additional Information

I. Gasparis
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: ioagaspa@math.okstate.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05594-5
Keywords: Ramsey theory, Schreier sets, dichotomy
Received by editor(s): February 19, 1999
Received by editor(s) in revised form: May 5, 1999
Published electronically: August 30, 2000
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

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