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 \({\mathfrak {p}}\). (shrink)

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers [Formula: see text] on [Formula: see text] as the prototype structures, we construct a class of continuum many topological Ramsey spaces [Formula: see text] which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudlák–Rödl theorem for barriers on the Ellentuck (...) space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text]. (shrink)