This question is motivated by discussion surrounding [this MO question](https://mathoverflow.net/questions/351827/a-new-cardinal-characteristic-related-to-partitions).


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 $\mathfrak{b}=\aleph\_1$? 



A question on simple $P_{\aleph_2}$-points
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](https://mathoverflow.net/questions/351827/a-new-cardinal-characteristic-related-to-partitions) 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**](http://dx.doi.org/10.1007/BF02764864), Isr. J. Math. 65, No. 3, 259-271 (1989). [ZBL0681.03033](https://zbmath.org/?q=an:0681.03033).


