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Weak partition properties on trees. (English) Zbl 1296.03024

The authors define a partition symbol for trees: \(\kappa\rightsquigarrow(\lambda)^{<\omega}_\omega\) means that if one colours the nodes of the trees \(K^{<\omega}\) with countably many colours then there is a \(\lambda\)-branching subtree such that the colouring takes on finitely many colours on each level (the \(\omega\) indicates the number of colours). The principle is related to a thinning-out method for such trees that the authors use to prove two results on metric spaces: (1) a Čech-analytic space is \(\sigma\)-locally compact iff it does not contain a closed copy of the space of irrational numbers, and (2) if a Čech-analytic metric space is not \(\sigma\)-locally separable then there is no sensible value for its Hausdorff dimension. The bulk of the paper is actually devoted to the partition relation and its generalizations. Intriguingly, the relation holds for regular \(\kappa\) below \(\mathfrak{p}\) and above \(\mathfrak{d}\), but not for \(\mathfrak{b}\) and \(\mathfrak{d}\) itself.
Reviewer: K. P. Hart (Delft)

MSC:

03E05 Other combinatorial set theory
03E15 Descriptive set theory
32C18 Topology of analytic spaces
54E40 Special maps on metric spaces
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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