Thin ultrafilters, P-hierarchu and MArtin Axiom

Under MA we prove that for the ideal $\cal I$ of thin sets on $ω$ and for any ordinal $γ\leq ω_1$ there is an ${\cal I}$-ultrafilter (in the sense of Baumgartner), which belongs to the class ${\cal P}_γ$ 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 Flašková, 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 $ω$ there is an ${\cal I}$-ultrafilter, however the ideal of thin sets is not P-ideal.

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