Countable dense homogeneity and $\lambda$-sets

Abstract

We show that all sufficiently nice $\lambda$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $\kappa \le \mathfrak{b}$ there is a countable dense homogeneous metric space of size $\kappa$. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size $\kappa$ is equivalent to the existence of a $\lambda$-set of size $\kappa$. On the other hand, it is consistent with the continuum arbitrarily large that every $\mathsf{CDH}$ metric space has size either $\omega_1$ or size $\mathfrak c$. An example of a Baire $\mathsf{CDH}$ metric space which is not completely metrizable is presented. Finally, answering a question of Arhangel'skii and van Mill we show that that there is a compact non-metrizable $\mathsf{CDH}$ space in ZFC.