  This paper investigates properties of $\sigma$-closed forcings which generate
ultrafilters satisfying weak partition relations. The Ramsey degree of an
ultrafilter $\mathcal{U}$ for $n$-tuples, denoted $t(\mathcal{U},n)$, is the
smallest number $t$ such that given any $l\ge 2$ and coloring
$c:[\omega]^n\rightarrow l$, there is a member $X\in\mathcal{U}$ such that the
restriction of $c$ to $[X]^n$ has no more than $t$ colors. Many well-known
$\sigma$-closed forcings are known to generate ultrafilters with finite Ramsey
degrees, but finding the precise degrees can sometimes prove elusive or quite
involved, at best. In this paper, we utilize methods of topological Ramsey
spaces to calculate Ramsey degrees of several classes of ultrafilters generated
by $\sigma$-closed forcings. These include a hierarchy of forcings due to
Laflamme which generate weakly Ramsey and weaker rapid p-points, forcings of
Baumgartner and Taylor and of Blass and generalizations, and the collection of
non-p-points generated by the forcings
$\mathcal{P}(\omega^k)/\mathrm{Fin}^{\otimes k}$. We provide a general approach
to calculating the Ramsey degrees of these ultrafilters, obtaining new results
as well as streamlined proofs of previously known results. In the second half
of the paper, we calculate pseudointersection and tower numbers for these
$\sigma$-closed forcings and their relationships with the classical
pseudointersection number $\mathfrak{p}$.
