Abstract
A P-point ultrafilter over \(\omega \) is called an interval P-point if for every function from \(\omega \) to \(\omega \) there exists a set A in this ultrafilter such that the restriction of the function to A is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under \(\textsf{CH}\) or \(\textsf{MA}\). (2) We identify a cardinal invariant \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})\) such that every filter base of size less than continuum can be extended to an interval P-point if and only if \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}\). (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption \({\mathfrak {d}}={\mathfrak {c}}\) or \(\textbf{cov}({\mathcal {B}})={\mathfrak {c}}\).
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Notes
In some literature, a summable ideal \({\mathcal {I}}_h\) is defined without requiring that h be monotonic. We choose the current form for simplicity.
We define \({\mathcal {C}}_{\tiny {\hbox {int}}}\) by requiring that all sets A in it have finite y-segments \(A^n\), rather than finite x-segments, due to the fact that in the proof of Lemma 4.1. the function f, as a convention, maps a number from x-axis to a number in y-axis. For keeping consistency, we then introduce the ideals \({\mathcal {I}}_{\tiny {\hbox {selec}}}\) and \({\mathcal {I}}_{\tiny {\hbox {P}}}\) as the reflections of \({\mathcal {E}}\!{\mathcal {D}}\) and \(\hbox {Fin}\times \hbox {Fin}\), respectively, along the diagonal line \(y=x\).
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Jialiang He: Supported in part by a National Science Foundation of China Grant #11801386. Renling Jin: Supported in part by a collaboration research Grant #513023 from Simons Foundation. Shuguo Zhang: Supported in part by a National Science Foundation of China Grant #11771311.
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He, J., Jin, R. & Zhang, S. Generic existence of interval P-points. Arch. Math. Logic 62, 619–640 (2023). https://doi.org/10.1007/s00153-022-00853-3
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DOI: https://doi.org/10.1007/s00153-022-00853-3