Proceedings of the American Mathematical Society

Author(s): I. Gasparis
Journal: Proc. Amer. Math. Soc. 129 (2001), 759-764.
MSC (1991): Primary 46B03; Secondary 06A07, 03E02
Posted: August 30, 2000
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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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