  A countable poset is ultrahomogeneous if every isomorphism between its finite
subposets can be extended to an automorphism. The groups
$\operatorname{Aut}(A)$ of such posets $A$ have a natural topology in which
$\operatorname{Aut}(A)$ are Polish topological groups. We consider the problem
whether $\operatorname{Aut}(A)$ contains a dense free subgroup of two
generators. We show that if $A$ is ultrahomogeneous, then
$\operatorname{Aut}(A)$ contains such subgroup. Moreover, we characterize whose
countable ultrahomogeneous posets $A$ such that for each natural $m$, the set
of all cyclically dense elements $\bar{g}\in\operatorname{Aut}(A)^m$ for the
diagonal action is comeager in $\operatorname{Aut}(A)^m$. In our considerations
we strongly use the result of Schmerl which says that there are essentially
four types of countably infinite ultrahomogeneous posets.
