Ramsey ultrafilters and countable-to-one uniformization (Q330063)

It is well-known that a generic filter for the Boolean algebra \({\mathcal P}(\omega)/\text{fin}\) is a Ramsey ultrafilter. The authors give a proof of a result by \textit{S. Todorcevic} [Introduction to Ramsey space. Princeton, NJ: Princeton University Press (2010; Zbl 1205.05001)] that establishes a converse: in certain large-cardinal circumstances, every Ramsey ultrafilter is generic on \({\mathcal P}(\omega)/\mathrm{fin}\) over~\(L(\mathbb{R})\). The main result of the paper is that forcing with \({\mathcal P}(\omega)/\mathrm{fin}\) preserves the following choice-like statement: every relation on~\(\mathbb{R}\) with countable vertical sections contains a function with the same domain. There is also a proof of a result of Woodin that the statement follows from the conjunction of \(\mathrm{AD}\), \(\mathrm{DC}_\mathbb{R}\), and a coding condition for sets of reals.