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Ramsey families of subtrees of the dyadic tree

Authors:	Vassilis Kanellopoulos

Institution:	Department of Mathematics, National Technical University of Athens, Athens 15780, Greece

Abstract:	We show that for every rooted, finitely branching, pruned tree of height $\omega$ there exists a family $\mathcal{F}$ which consists of order isomorphic to subtrees of the dyadic tree $C=\{0,1\}^{<\mathbb{N} }$ with the following properties: (i) the family $\mathcal{F}$ is a $G_\delta$ subset of ; (ii) every perfect subtree of contains a member of $\mathcal{F}$ ; (iii) if is an analytic subset of $\mathcal{F}$ , then for every perfect subtree of there exists a perfect subtree of such that the set $\{A\in\mathcal{F}: A\subseteq S'\}$ either is contained in or is disjoint from .

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