The post is a bit long, but all the questions are similar or concern the same topic.
Let $\omega^*=\beta\omega\setminus\omega$. A well-known topological definition of a P-point (on $\omega$) is as follows: a point $x\in\omega^*$ is a P-point if every intersection of countably many open neighborhoods of $x$ contains an open neighborhood of $x$. Similarly, a point $x\in\omega^*$ is a weak P-point if $x$ is not in the closure of any countable subset not containing $x$. We have also an equivalent definition of P-points in terms of functions $\omega\to\omega$: an ultrafilter $x\in\omega^*$ is a P-point if for every function $f\colon\omega\to\omega$ there is $A\in x$ such that $f\restriction A$ is either constant or finite-to-one. Since every P-point is a weak P-point, my first question is as follows:
Question 1. Does there exist a similar characterization of weak P-points (i.e. in terms of functions $\omega\to\omega$)?
Kunen proved in ZFC that we always have $2^{\mathfrak{c}}$ weak P-points and at least $\mathfrak{c}$ incomparable in the sense of Rudin-Keisler weak P-points. My next questions concern the total number of (incomparable/incompatible) P-points provided that there is at least one P-point.
Question 2. Assume that a P-point exists. (a) Does there exist another (non-isomorphic or incomparable) P-point? (b) Do there exist $2^{\mathfrak{c}}$ different P-points? (c) Do there exist $\mathfrak{c}$ incomparable P-points?
(It is easy to see that there exist $\mathfrak{c}$ isomorphic P-points.)
Question 3. Assume that there exist $2^{\mathfrak{c}}$ many P-points. (a) Do we have then $2^{\mathfrak{c}}$ (or at least $\mathfrak{c})$ incomparable P-points? (b) Do there exist $2^{\mathfrak{c}}$ (or at least $\mathfrak{c}$, or even $2$) incompatible P-points?
Let us say that two ultrafilters $U,V\in\omega^*$ are near coherent if there exists a finite-to-one function $f\colon\omega\to\omega$ such that $f(U)=f(V)$. The near coherence is an equivalence relation, so we can count the number of the equivalence classes. The Near Coherence of Filters principle (NCF in short) states that there exists only one equivalence class. Blass and Shelah constructed a model of set theory in which the NCF holds (it is now also known to hold in the Miller model). On the other hand, Banakh and Blass proved that either we have finitely many equivalence classes or $2^{\mathfrak{c}}$ (the latter holds e.g. in each model where $\mathfrak{u}\ge\mathfrak{d}$, so e.g. under CH). The next question is in the same spirit as Questions 2 and 3.
Question 4. Assume that we have $2^{\mathfrak{c}}$ many near coherence classes. Does there exist $\mathfrak{c}$ (or $2^{\mathfrak{c}}$) many incompatible in the sense of Rudin-Keisler ordering weak P-points?
It is believed (but not proved so far) that there exists a model with exactly $2$ near coherence classes and for every $n>2$ there is no model with $n$ classes. Also, the NCF implies the existence of a P-point. Thus, my next (and last) question is the following.
Question 5. Assume there are exactly 2 classes of near coherence. Does there exist any P-point? If yes, then are there two that are not compatible?
Thank you very much for the help!
