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

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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 χF 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.