Given an ultrafilter \(\mathcal U\) on \(\omega\), the Ramsey degree of \(\mathcal U\) for \(n\)-tuples, denoted \(t(\mathcal U,n)\), is the smallest number \(t\) (if some such number exists) such that given any \(\ell \geq 2\) and any coloring \(c: [\omega]^n \rightarrow \ell\), there is some \(X \in \mathcal U\) such that the restriction of \(c\) to \([X]^n\) has \(\leq\!t\) colors. For example, \(\mathcal U\) is a Ramsey ultrafilter if and only if \(t(\mathcal U,n) = 1\) for all \(n\), and \(\mathcal U\) is a weakly Ramsey ultrafilter if \(t(\mathcal U,2) = 2\).
A number of \(\sigma\)-closed posets can be used to add ultrafilters with finite Ramsey degrees. The most well-known example of this is that forcing with \(\mathcal P(\omega) / \mathrm{Fin}\) adds a Ramsey ultrafilter, i.e., an ultrafilter with Ramsey degree \(t(\mathcal U,n) = 1\) for every \(n\). While many other \(\sigma\)-closed forcings are known to generate ultrafilters with finite Ramsey degrees, the precise value of these degrees is often difficult to determine, and had remained unknown in many cases. In this paper, the authors present a general method for calculating these numbers using the tools of topological Ramsey spaces. They are able to determine the Ramsey degrees of several classes of ultrafilters generated by \(\sigma\)-closed forcings whose degrees were not previously known.
Later in the paper, the authors also investigate the pseudointersection and tower numbers for these same \(\sigma\)-closed forcings, emphasizing the relationship of these numbers with the classical cardinal characteristic \(\mathfrak{p}\). It remains an interesting open question whether the pseudointersection and tower numbers associated to a given topological Ramsey space must be equal to each other.