Weak partition properties on trees

Abstract

We investigate the following weak Ramsey property of a cardinal ?: If ? is coloring of nodes of the tree ? \mathfrak{d}}$$ is regular, then $${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$$ and that $${\mathfrak{b}}$$ $${(\mathfrak{b})^{ < \omega}_{\omega}}$$ and $${\mathfrak{d}}$$ $${(\mathfrak{d})^{ < \omega}_{\omega}}$$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A ?ech-analytic space is ?-locally compact iff it does not contain a closed homeomorphic copy of irrationals.