A new cardinal characteristic (related to partitions)?

In this post I will discuss some cardinal characteristic of the continuum, related to partitions of $$\omega$$ and would like to know if it is equal to some known cardinal characteristic.

By a partition of $$\omega$$ I understand a cover of $$\omega$$ by pairwise disjoint nonempty subsets. A partition $$\mathcal P$$ is called finitary if $$\sup_{P\in\mathcal P}|P|$$ is finite.

A family $$\mathfrak P$$ of partitions of $$\omega$$ is called directed if for any two partitions $$\mathcal A,\mathcal B\in\mathfrak P$$ there exists a partition $$\mathcal C\in\mathfrak P$$ such that each set $$S\in\mathcal A\cup\mathcal B$$ is contained in some set $$C\in\mathcal C$$.

Let $$\mathfrak P$$ is a family of partitions of $$\omega$$. An infinite subset $$D\subset\omega$$ is called $$\mathfrak P$$-discrete if for any partition $$\mathcal P\in\mathfrak P$$ there exists a finite set $$F\subset D$$ such that for any $$P\in\mathcal P$$ the intersection $$P\cap (D\setminus F)$$ contains at most one point.

Let $$\kappa$$ be the smallest cardinality of a directed family $$\mathfrak P$$ of finitary partitions of $$\omega$$ admitting no infinite $$\mathfrak P$$-discrete set $$D\subset\omega$$.

It can be shown that $$\mathfrak b\le\kappa\le\mathfrak c$$ (the upper bound follows from the observation that any maximal directed family of finitary partitions has no infinite discrete set, see Proposition 6.5 in this preprint).

Problem 1. Is $$\kappa$$ equal to some known cardinal characteristic of the continuum?

Problem 2. Is $$\kappa=\mathfrak c$$ in ZFC?

Problem 3. Find lower and upper bounds on $$\kappa$$ (which are better than $$\mathfrak b\le\kappa\le\mathfrak c$$).

Added in Edit. The lower bound $$\sup_{U\in\beta\omega}\pi(U)\le\kappa$$, suggested by Todd Eisworth can be improved to $$\mathfrak s\le \kappa$$. One can also prove that $$\max\{\mathfrak b,\mathfrak s,\}\le\mathfrak j\le\kappa\le\mathrm{non}(\mathcal M)$$ and hence $$\kappa$$ is not equal to $$\mathfrak c$$. The cardinal $$\mathfrak j$$ is discussed in this MO-post.

• Possibly a stupid sub-question. For any finitary partition ${\cal P}$ let $\mu({\cal P})$ be the smallest $k\in \omega$ such that for all $P\in{\cal P}$ we have $|P|\leq k$. Let ${\frak P}$ be a directed family of finitary partitions of $\omega$ with the property that there is $N\in \omega$ with $\mu({\cal P}) \leq N$ for all ${\cal P}\in {\frak P}$. (In other words, there is a global "block size bound" for every partition in ${\frak P}$.) Does this imply that ${\frak P}$ admint a ${\frak P}$-discrete set $D\in[\omega]^\omega$? Feb 6, 2020 at 20:28
• @DominicvanderZypen Very good question! The answer is "yes". If $\mathfrak P$ is directed and the cardinality of cells is bounded from above, then there exists an increasing sequence of partitions $(\mathcal P_n)_{n\in\omega}$ in $\mathfrak P$ that converges (in a pointwise sense) to some partition $\mathcal P_\infty$. Take a $\{\mathcal P_\infty\}$-discrete set and notice that it is also $\mathfrak P$-discrete. Feb 6, 2020 at 21:43
• Thanks for your answer, Taras! Feb 7, 2020 at 13:05

This is not an answer, but hopefully it's a helpful observation:

(1) If $$U$$ is an ultrafilter on $$\omega$$ and $$\mathcal{P}$$ is a finitary partition of $$\omega$$, then there is $$A\in U$$ such that $$A\cap P$$ contains at most one element for each $$P\in\mathcal{P}$$.

(As if each piece of the partition has cardinality at most $$n$$, then there is a $$k\leq n$$ such that the union of pieces with size exactly $$k$$ is in $$U$$. Now split this union up into $$k$$ pieces in the obvious way, and one of these is in $$U$$.)

(2) Given an ultrafilter $$U$$, let $$\tau(U)$$ be the least cardinal $$\tau$$ such that some subfamily of $$U$$ of cardinality $$\tau$$ fails to have an infinite pseudo-intersection. (We do not require the pseudo-intersection to be in $$U$$, so $$\aleph_1\leq\tau(U)\leq\mathfrak{c}$$.)

Observation:
If $$U$$ is an ultrafilter on $$\omega$$, then $$\tau(U)\leq\kappa$$.

Proof. Given a family $$\mathfrak{P}$$ of finitary partitions of $$\omega$$ (directed or not), we fix for each $$P\in\mathfrak{P}$$ a set $$A_P\in U$$ meeting each element of $$P$$ in at most one point. If $$|\mathfrak{P}|<\tau(U)$$ then we can find an infinite pseudo-intersection $$X$$ for the collection $$\{A_P:P\in\mathfrak{P}\}$$, and $$X$$ is $$\mathfrak{P}$$-discrete.$$_\square$$

I don't know anything about the cardinals $$\tau(U)$$. I note that at one point Blass and Shelah claimed to have model containing both simple $$P_{\aleph_1}$$ and simple $$P_{\aleph_2}$$ points, but Alan Dow discovered an error in the paper, and I'm not sure if it has ever been repaired. (The existence of a simple $$P_{\aleph_1}$$-point implies $$\mathfrak{b}=\mathfrak{u}=\aleph_1$$, while the simple $$P_{\aleph_2}$$ point is an ultrafilter $$U$$ with $$\tau(U)=\aleph_2$$. In such a model, $$\kappa$$ would be strictly greater than $$\mathfrak{b}$$.)

Clearly this is all tied up with the topology of $$\beta\omega$$, so I suspect much more is known by the experts.

• Indeed, very interesting! Especially the cardinal $\sup_{U\in\omega^*}\tau(U)$. Maybe @Andreas Blass could comment on the consistency of $\mathfrak b<\sup_{U\in\omega^*}\tau(U)$? Feb 3, 2020 at 20:13
• And what about the consistency of $\sup_{U\in\omega^*}\tau(U)<\mathfrak b$? Is there any model of ZFC where this strict inequality is true? Feb 3, 2020 at 20:20
• As far as I know, the error about simple P-points for two different cardinals has not yet been repaired. Shelah and Mildenberger each had arguments to repair it, but neither argument seems to have survived. Feb 3, 2020 at 22:57
• I think the second comment by @TarasBanakh can be answered by starting with a model of MA + $\neg$CH and then adjoining $\aleph_1$ random reals. The random reals don't affect $\mathfrak b$, which stays large. But any ultrafilter will have to contain each of the random reals or its complement, and I don't see any chance for a pseudo-intersection (which would have to depend on only countably many of the random reals, by ccc). (Unfortunately, I have no answer yet for the first comment, the one relevant to the actual question.) Feb 3, 2020 at 23:02
• It is consistent that $\mathfrak{b}<\kappa$. See the answer to mathoverflow.net/questions/352034/… Feb 14, 2020 at 0:56