The geometry of fractal sets
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We investigate the following weak Ramsey property of a cardinal 驴: If 驴 is coloring of nodes of the tree 驴 by countably many colors, call a tree $${T \subseteq \kappa^{ 驴-homogeneous if the number of colors on each level of T is finite. Write $${\kappa \rightsquigarrow (\lambda)^{ to denote that for any such coloring there is a 驴-homogeneous 驴-branching tree of height 驴. We prove, e.g., that if $${\kappa or $${\kappa \mathfrak{d}}$$ is regular, then $${{\kappa \rightsquigarrow (\kappa)^{ and that $${\mathfrak{b}}$$ $${(\mathfrak{b})^{ and $${\mathfrak{d}}$$ $${(\mathfrak{d})^{ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A 驴ech-analytic space is 驴-locally compact iff it does not contain a closed homeomorphic copy of irrationals.