Weak partition properties on trees

Abstract

We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ ^<ω by countably many colors, call a tree ${T \subseteq \kappa^{ < \omega}}$ χ-homogeneous if the number of colors on each level of T is finite. Write ${\kappa \rightsquigarrow (\lambda)^{ < \omega}_{\omega}}$ to denote that for any such coloring there is a χ-homogeneous λ-branching tree of height ω. We prove, e.g., that if ${\kappa < \mathfrak{p}}$ or ${\kappa > \mathfrak{d}}$ 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 Čech-analytic space is σ-locally compact iff it does not contain a closed homeomorphic copy of irrationals.