  In this paper we start the analysis of the class $\mathcal D_{\aleph_2}$, the
class of cofinal types of directed sets of cofinality at most $\aleph_2$. We
compare elements of $\mathcal D_{\aleph_2}$ using the notion of Tukey
reducibility. We isolate some simple cofinal types in $\mathcal D_{\aleph_2}$,
and then proceed to show which of these types have an immediate successor in
the Tukey ordering of $\mathcal D_{\aleph_2}$.
