Ramsey families of subtrees of the dyadic tree

Kanellopoulos, Vassilis

doi:10.1090/S0002-9947-05-03968-1

Ramsey families of subtrees of the dyadic tree
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Trans. Amer. Math. Soc. 357 (2005), 3865-3886 Request permission

Abstract:

We show that for every rooted, finitely branching, pruned tree $T$ of height $\omega$ there exists a family $\mathcal {F}$ which consists of order isomorphic to $T$ 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 $2^C$; (ii) every perfect subtree of $C$ contains a member of $\mathcal {F}$; (iii) if $K$ is an analytic subset of $\mathcal {F}$, then for every perfect subtree $S$ of $C$ there exists a perfect subtree $S’$ of $S$ such that the set $\{A\in \mathcal {F}: A\subseteq S’\}$ either is contained in or is disjoint from $K$.

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