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

A Topological View of Ramsey Families of Finite Subsets of Positive Integers 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

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Publisher
Birkhäuser-Verlag
Copyright
Copyright © 2007 by Birkhäuser Verlag, Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-006-2043-9
Publisher site
See Article on Publisher Site

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.

Journal

PositivitySpringer Journals

Published: Jan 1, 2006

References

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