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.