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

Gasparis, I.

doi:10.1090/S0002-9939-00-05594-5

A dichotomy theorem for subsets of the power set of the natural numbers
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by I. Gasparis

Proc. Amer. Math. Soc. 129 (2001), 759-764

DOI: https://doi.org/10.1090/S0002-9939-00-05594-5

Published electronically: August 30, 2000

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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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