A question on simple $P_{\aleph_2}$-points

Question

This question is motivated by discussion surrounding this MO question.

An ultrafilter $U$ on $\omega$ is a simple $P_{\aleph_2}$-point if it is generated by a sequence $\langle X_\alpha:\alpha<\omega_2\rangle$ such that

$$\alpha<\beta\Longrightarrow |X_\beta\setminus X_\alpha|<\aleph_0,$$

that is, generated by an $\omega_2$ sequence of subsets of $\omega$ that is decreasing modulo the ideal of finite sets.

Question

Is the existence of a simple $P_{\aleph_2}$-point consistent with $\mathfrak{b}=\aleph_1$?

Answer 1

The answer is "yes". Alan Dow pointed out in correspondence that the construction of Blass and Shelah referenced below can be modified to yield such a model. The particular model arises by using finite support to add an $\omega_1$-sequence of Cohen reals, followed by a finite support iteration adding the generating set for the $P_{\aleph_2}$-point using a variant of Mathias forcing. The salient point of the construction is that the Mathias-type forcing (really Mathias forcing with respect to a carefully chosen ultrafilter at each stage) does not add a real dominating the Cohen reals, and so $\mathfrak{b}=\aleph_1$ in the extension.

Referring to the discussion on the question of Banakh that motivated our question, this shows there is a model in which Banakh's cardinal $\kappa$ is strictly greater than $\mathfrak{b}$, as the $P_{\aleph_2}$-point is an ultrafilter $U$ with $\tau(U)=\aleph_2$.

Blass, Andreas; Shelah, Saharon, Ultrafilters with small generating sets, Isr. J. Math. 65, No. 3, 259-271 (1989). ZBL0681.03033.