# Pseudo-intersections, splitting families, and ultrafilters

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, Ann. Pure Appl. Logic 33, 213-243 (1987). ZBL0634.03047.

• Is it true that $\tau(U)$ is a regular cardinal, for every non-principal ultrafilter $U$ on $\omega$? – Santi Spadaro Feb 16 '20 at 23:22
• I don't know the answer to that one. A priori, I don't see why this would need to occur, but I have not thought deeply about it. – Todd Eisworth Feb 17 '20 at 1:11
• I think another good candidate for a counterexample could be the Mathias model. – Will Brian Feb 17 '20 at 17:58
• I guess it’s really about reflection dealing with questions like “If every small subfamily has a pseudointersection, does the whole family?” – Todd Eisworth Feb 17 '20 at 19:19

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:

Jörg Brendle and Saharon Shelah, Ultrafilters on $$\omega$$ -- their ideals and their characteristics,'' Transactions of the AMS 351 (1999), pp. 2643-2674. (available here)

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:

Alan Dow and Saharon Shelah, Pseudo P-points and splitting number,'' Archive for Mathematical Logic 58 (2019), pp. 1005-10027. (available here)

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 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.

• Perfect! And thank you for the references! – Todd Eisworth Feb 17 '20 at 22:36
• This can be modified to push up $\mathfrak{h}$ as well by sprinkling in more Mathias forcing on a stationary/co-stationary set of ordinals of cofinality $\omega_1$. Can the Dow-Shelah construction accomplish the same? – Todd Eisworth Feb 18 '20 at 3:04