Quasi-selective and weakly Ramsey ultrafilters

Selective ultrafilters are characterized by many equivalent properties, in particular the Ramsey property that every finite colouring of unordered pairs of integers has a homogeneous set in U, and the equivalent property that every function is nondecreasing on some set in U. Natural weakenings of these properties led to the inequivalent notions of weakly Ramsey and of quasi-selective ultrafilter, introduced and studied in [1] and [4], respectively. U is weakly Ramsey if for every finite colouring of unordered pairs of integers there is a set in U whose pairs share only two colours, while U is f-quasi-selective if every function g < f is nondecreasing on some set in U. (So the quasi-selective ultrafilters of [4] are here id-quasi selective.) In this paper we consider the relations between various natural cuts of the ultrapowers of N modulo weakly Ramsey and f-quasi-selective ultrafilters. In particular we characterize those weakly Ramsey ultrafilters that are isomorphic to a quasi-selective ultrafilter.