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Weak partition properties on trees

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Abstract

We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ 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.

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Correspondence to Ondřej Zindulka.

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Work on this project was conducted during the third author’s sabbatical stay at the Instituto de matemáticas, Unidad Morelia, Universidad Nacional Autonóma de México supported by CONACyT grant no. 125108. The third author was also supported by The Czech Republic Ministry of Education, Youth and Sport, research project BA MSM 210000010. The first author gratefully acknowledges support from PAPIIT grant IN101608 and CONACYT grant 80355.

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Hrušák, M., Simon, P. & Zindulka, O. Weak partition properties on trees. Arch. Math. Logic 52, 543–567 (2013). https://doi.org/10.1007/s00153-013-0331-1

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