  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.
