Countable Dense Homogeneity

We prove that a connected, countable dense homogeneous space is $n$-homogeneous for every $n$, and strongly 2-homogeneous provided it is locally connected. We also present an example of a connected and countable dense homogeneous space which is not strongly 2-homogeneous. We prove that a countable dense homogeneous space has size at most continuum. If it moreover is compact, then it is first-countable under the Continuum Hypothesis. We also construct under the Continuum Hypothesis an example of a hereditarily separable, hereditarily Lindelöf, countable dense homogeneous compact space of uncountable weight. We also discuss locally compact separable metrizable spaces with a finite number of types of countable dense sets and prove a structure theorem for them. Some of the presented results were obtained with A.V. Arhangelskii and M. Hrusak.