Type: Article

Weak partition properties on trees

Hrušák, Michael; Simon, Petr; Zindulka, Ondrej;

Abstract:

We investigate the following weak Ramsey property of a cardinal.: If. is coloring of nodes of the tree kappa(<omega) by countably many colors, call a tree T subset of kappa(<omega) chi-homogeneous if the number of colors on each level of T is finite. Write kappa (sic) (lambda)(omega)(<omega) to denote that for any such coloring there is a chi-homogeneous lambda-branching tree of height omega. We prove, e.g., that if kappa < p or kappa > partial derivative is regular, then kappa (sic) (kappa)(omega)(<omega) and that b (sic) ( b)(omega)(<omega) and partial derivative (sic) (partial derivative)(omega)(<omega) . The arrow is applied to prove a generalization of a theorem of Hurewicz: A C. ech- analytic space is s- locally compact iff it does not contain a closed homeomorphic copy of irrationals.