Characterizing existence of certain ultrafilters

Following Baumgartner (1995) , for an ideal on , we say that an ultrafilter on is an if for every function there is with .If there is an -ultrafilter which is not a -ultrafilter, then is not below in the Katětov order (i.e. for every function there is with ). On the other hand, in general does not imply that existence of an -ultrafilter which is not a -ultrafilter is consistent.We provide some sufficient conditions on ideals to obtain the equivalence: if and only if it is consistent that there exists an -ultrafilter which is not a -ultrafilter. In some cases when the Katětov order is not enough for the above equivalence, we provide other conditions for which a similar equivalence holds. We are mainly interested in the cases when the family of all -ultrafilters or -ultrafilters coincides with some known family of ultrafilters: P-points, Q-points or selective ultrafilters (a.k.a. Ramsey ultrafilters). In particular, our results provide a characterization of Borel ideals which can be used to characterize P-points as -ultrafilters.Moreover, we introduce a cardinal invariant which is used to obtain a sufficient condition for the existence of an -ultrafilter which is not a -ultrafilter. Finally, we prove some new results concerning existence of certain ultrafilters under various set-theoretic assumptions.