# 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 $D\in[\omega]^\omega$? – Dominic van der Zypen Feb 6 '20 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. – Taras Banakh Feb 6 '20 at 21:43
• Thanks for your answer, Taras! – Dominic van der Zypen Feb 7 '20 at 13:05

(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 $\mathfrak b<\sup_{U\in\omega^*}\tau(U)$? – Taras Banakh Feb 3 '20 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? – Taras Banakh Feb 3 '20 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. – Andreas Blass Feb 3 '20 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.) – Andreas Blass Feb 3 '20 at 23:02
• It is consistent that $\mathfrak{b}<\kappa$. See the answer to mathoverflow.net/questions/352034/… – Todd Eisworth Feb 14 '20 at 0:56