Suppose $U$ is a non-principal ultrafilter on $\omega$, and let us define $\tau(U)$ to be the minimum cardinality of a family $\mathcal{X}\subseteq U$ such that $\mathcal{X}$ does not have an infinite pseudo-intersection, that is, there is no infinite $A$ such that $A\setminus B$ is finite for all $B\in \mathcal{X}$.


**Claim**: $\tau(U)\leq\mathfrak{s}$ for any $U$. 


The point is that $U$ will contain a splitting family of cardinality $\mathfrak{s}$, and this splitting family has no infinite pseudo-intersection. (To see the first statement, note that if $\mathcal{X}$ is a splitting family and we replace some of the elements of $\mathcal{X}$ by their complements, then the resulting collection is still a splitting family. Since an ultrafilter will contain one of $X$ and $\omega\setminus X$ for each $X\in\mathcal{X}$, we may as well assume $\mathcal{X}\subseteq U$.)


**Question**: Is there (in ZFC) an ultrafilter $U$ on $\omega$ for which $\tau(U)=\mathfrak{s}$?


I conjecture that the answer is "no", and that this negative answer will be witnessed in the original Blass-Shelah model of NCF from the paper below. 


*Blass, Andreas; Shelah, Saharon*, [**There may be simple $P\_{\aleph \_ 1}$- and $P\_{\aleph \_ 2}$-points and the Rudin-Keisler ordering may be downward directed**](http://dx.doi.org/10.1016/0168-0072(87)90082-0), Ann. Pure Appl. Logic 33, 213-243 (1987). [ZBL0634.03047](https://zbmath.org/?q=an:0634.03047).



Pseudo-intersections, splitting families, and ultrafilters
The answer is no -- it is consistent that every $U \in \omega^\*$ has $\tau(U) < \mathfrak{s}$.


I had an idea for proving this earlier today, using the Mathias model. I couldn't quite make things work, and I ended up talking about the problem with Alan Dow for a good part of the afternoon. (1) We still think the Mathias model could work, but it seems tricky. (2) There's a fix: if you interleave Laver forcings with Mathias forcings in a certain way, then the resulting iteration does work. (I'll sketch this below.) (3) I learned from Alan that your question has been studied already. The consistency of "every $U \in \omega^\*$ has $\tau(U) < \mathfrak{s}$" is already known (via a different argument than the Mathias-Laver iteration sketched below), and the characteristic $\tau(U)$ has been studied quite a bit.


A rich source of information on $\tau(U)$ is the following paper by Brendle and Shelah:



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> Jörg Brendle and Saharon Shelah, ``Ultrafilters on $\omega$ -- their ideals and their characteristics,'' *Transactions of the AMS* **351** (1999), pp. 2643-2674. (available [here](https://arxiv.org/pdf/math/9710217.pdf))
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What you call $\tau(U)$ is in this paper called $\pi \mathfrak{p}(U)$. They prove, among other things, that


$\bullet$ If $U$ is not a $P$-point, then $\tau(U) \leq \mathfrak{b}.$ 


$\bullet$ If $U$ is not a $P$-point, then the cofinality of $\tau(U) \leq \mathfrak{b}$ is uncountable. (This provides a partial answer to Santi's question in the comments.)


$\bullet$ A characterization of $\tau(U)$ is given in terms of an ideal defined from Ramsey-null sets.


$\bullet$ It is consistent that $\tau(U) < \mathfrak{s}$ for all $U \in \omega^\*$.


The first two results are in Section 2, the next in Section 3, and the last in Section 7. The last result answers your question, of course, but I should also mention another relevant paper:



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> Alan Dow and Saharon Shelah, ``Pseudo P-points and splitting number,'' *Archive for Mathematical Logic* **58** (2019), pp. 1005-10027. (available [here](https://webpages.uncc.edu/~adow/F1730.pdf))
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In the Brendle-Shelah paper, they prove $\sup\_{U \in \omega^\*}\tau(U) < \mathfrak{s}$ is consistent, but the gap between $\sup\_{U \in \omega^\*}\tau(U)$ and $\mathfrak{s}$ is only one. In the Dow-Shelah paper, they use a more complicated matrix iteration to make the gap between $\sup\_{U \in \omega^\*}\tau(U)$ and $\mathfrak{s}$ arbitrarily large.


Finally, let me sketch the idea I mentioned above. The idea is to do a countable support iteration that uses Laver forcing at limit steps of cofinality $\omega\_1$, and uses Mathias forcing everywhere else. The iteration is of length $\omega\_2$ and CH holds in the ground model. Let $V[G]$ denote the result of such an iteration, and let $U \in \omega^\*$ in $V[G]$. By reflection, there is some intermediate model $V[G\_\alpha]$ with $\alpha < \omega\_2$ where $U \cap V[G\_\alpha]$ is an ultrafilter in $V[G\_\alpha]$, and where $\alpha$ has cofinality $\omega\_1$. At this stage, we force with Laver forcing, and this adds a length-$\omega\_1$ tower to $U \cap V[G\_\alpha]$. All the subsequent Laver forcings and Mathias forcings preserve the fact that this tower has no pseudo-intersection, and so this tower is a witness to the fact that $\tau(U) = \aleph\_1$ in the final extension. (See Theorem 7.11 from [this](https://webpages.uncc.edu/adow/cohenRandom.pdf) paper of Alan's for more detail on the last two sentences.) Finally, the Mathias forcings enable us to get $\mathfrak{s} = \mathfrak{c}$. This is well-known to be true in the Mathias model (although not in the Laver model), and it's true in this model for essentially the same reasons. 


