How high can Baumgartner's {\cal I}-ultrafilters lie in the P-hierarchy?

TL;DR

Under the Continuum Hypothesis, the paper constructs ${ m I}$-ultrafilters within the P-hierarchy of ultrafilters, demonstrating the potential height of such ultrafilters beyond P-points.

Contribution

The paper proves the existence of ${ m I}$-ultrafilters at any level of the P-hierarchy, extending previous results about P-points.

Findings

Existence of ${ m I}$-ultrafilters in all classes ${ m P}_eta$ for $eta o \omega_1$

Generalization of Flašková's theorem on ${ m I}$-ultrafilters not being P-points

Construction under CH of ultrafilters with high P-hierarchy rank

Abstract

Under CH we prove that for any tall ideal $\cal I$ on $\omega$ and for any ordinal $\gamma \leq \omega_1$ there is an ${\cal I}$-ultrafilter (in the sense of Baumgartner), which belongs to the class ${\cal P}_{\gamma}$ of P-hierarchy of ultrafilters. Since the class of ${\cal P}_2$ ultrafilters coincides with a class of P-points, out result generalize theorem of Fla\v{s}kov\'a, which states that there are ${\cal I}$-ultrafilters which are not P-points.