Following Baumgartner [J. Symb. Log. 60 (1995), no. 2], for an ideal $\mathcal{ I}$ on $\omega$, we say that an ultrafilter $\mathcal{ U}$ on $\omega$ is an $\mathcal{ I}$-ultrafilter if for every function $f:\omega\to\omega$ there is $A\in \mathcal{ U}$ with $f[A]\in \mathcal{ I}$. If there is an $\mathcal{ I}$-ultrafilter which is not a $\mathcal{ J}$-ultrafilter, then $\mathcal{ I}$ is not below $\mathcal{ J}$ in the Kat\v{ e}tov order $\leq_{ K}$ (i.e. for every function $f:\omega\to\omega$ there is $A\in \mathcal{ I}$ with $f^{ -1}[A]\notin \mathcal{ J}$). On the other hand, in general $\mathcal{ I}\not\leq_{ K}\mathcal{ J}$ does not imply that existence of an $\mathcal{ I}$-ultrafilter which is not a $\mathcal{ J}$-ultrafilter is consistent. We provide some sufficient conditions on ideals to obtain the equivalence: $\mathcal{ I}\not\leq_{ K}\mathcal{ J}$ if and only if it is consistent that there exists an $\mathcal{ I}$-ultrafilter which is not a $\mathcal{ J}$-ultrafilter. In some cases when the Kat\v{ e}tov 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 $\mathcal{ I}$-ultrafilters or $\mathcal{ J}$-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 $\mathcal{ I}$ which can be used to characterize P-points as $\mathcal{ I}$-ultrafilters. Moreover, we introduce a cardinal invariant which is used to obtain a sufficient condition for the existence of an $\mathcal{ I}$-ultrafilter which is not a $\mathcal{ I}$-ultrafilter. Finally, we prove some new results concerning existence of certain ultrafilters under various set-theoretic assumptions.
Categories: math.LO