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A Topological View of Ramsey Families of Finite Subsets of Positive Integers

Authors:

Persephone Kiriakouli Nikolaos Papanastassiou

Institution:

(1) Mpiskini 29 Zografou, 15771 Athens, Greece;(2) Department of Mathematics, University of Athens, Panepistimiopolis, 15784 Athens, Greece

Abstract:

If $\mathcal F$ is an initially hereditary family of finite subsets of positive integers (i.e., if $F \in \mathcal F$ and G is initial segment of F then $G \in \mathcal F$ ) and M an infinite subset of positive integers then we define an ordinal index $\alpha_{M}( \mathcal F )$ . We prove that if $\mathcal F$ is a family of finite subsets of positive integers such that for every $F \in \mathcal F$ the characteristic function χ_F is isolated point of the subspace

$X_{\mathcal F}= \{ \chi_{G}: G \mbox{ is initial segment of $F$ for some } F \in \mathcal F \}$

of { 0,1 }^N with the product topology then $\alpha_{M}( \overline{\mathcal F} )< \omega_{1}$ for every $M \subseteq {\rm N}$ infinite, where $\overline{\mathcal F}$ is the set of all initial segments of the members of $\mathcal F$ and ω₁ is the first uncountable ordinal. As a consequence of this result we prove that $\mathcal F$ is Ramsey, i.e., if $\{ {\mathcal P}_{1}, {\mathcal P}_{2} \}$ is a partition of $\mathcal F$ then there exists an infinite subset M of positive integers such that

$\mathcal F \cap M]^{< \omega} \subseteq {\mathcal P}_{1} \quad \mbox{or} \quad \mathcal F \cap M]^{< \omega} \subseteq {\mathcal P}_{2},$

where M]^{< ω} is the family of all finite subsets of M.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 05A17 05A18

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