Thin ultrafilters, P-hierarchu and MArtin Axiom

TL;DR

Under Martin's Axiom, the paper proves the existence of certain ultrafilters related to thin sets and P-hierarchy, extending previous results and connecting to the structure of ultrafilters under different set-theoretic assumptions.

Contribution

The paper establishes the existence of ${ m I}$-ultrafilters within the P-hierarchy for the ideal of thin sets under MA, generalizing known results about P-points.

Findings

Existence of ${ m I}$-ultrafilters in ${ m P}_eta$ classes under MA.

Extension of Flašková's theorem to thin set ideals.

Connection between thin sets, P-hierarchy, and set-theoretic axioms.

Abstract

Under MA we prove that for the ideal $\cal I$ of thin sets 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. It is also related to theorem which states that under CH for any tall P-ideal $\cal I$ on $\omega$ there is an ${\cal I}$-ultrafilter, however the ideal of thin sets is not P-ideal.