Countable dense homogeneity of definable spaces
  We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a Borel X subset of 2^omega the following are equivalent: (1) X is G_delta in 2^omega, (2) X^omega is CDH and (3) X^omega is homeomorphic to 2^omega or to omega^omega. Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Steprans and Zhou by showing that the cardinal p = min{kappa: 2^kappa is not CDH}. 
Michael Hrusak, Beatriz Zamora Aviles