Thinning directed sets ${\frak P}$ of partitions of $\omega$ with no ${\frak P}$-discrete subsets

Question

This question branches from Taras Banakh's recent question on a cardinal characteristic connected to families of partitions that are directed in the ordering of partition refinement.

A partition $\mathcal P$ of $\omega$ is said to be finitary if there is $k\in \omega$ such that for every $P\in {\cal P}$ we have $|P|\leq k$.

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 ${\frak P}$ be a directed family of finitary partitions admitting no ${\frak P}$-discrete set. Is there ${\frak C}\subseteq {\frak P}$ with the following properties?

1. ${\frak C}$ admits no ${\frak C}$-discrete subset, and
2. for all ${\cal C}_1, {\cal C}_2\in {\frak C}$ we have that either ${\cal C}_1$ refines ${\cal C}_2$, or vice versa. (If ${\cal P}, {\cal Q}$ are partitions of $\omega$, we say that ${\cal P}$ refines ${\cal Q}$ if every member of ${\cal P}$ is contained in some (i.e. exactly one) member of ${\cal Q}$.)

Answer 1

The answer to this question is negative.

Given a linearly ordered family $\mathfrak C$ of finitary partitions of $\omega$, write $\mathfrak C=\bigcup_{n=1}^\infty\mathfrak C_n$ where $\mathfrak C_n=\{\mathcal P\in\mathfrak C:\sup_{P\in\mathcal P}|P|\le n\}$.

For a partition $\mathcal P$ of $\omega$ and a point $x\in\omega$, let $\mathcal P(x)$ be the unique set in the partition $\mathcal P_n$ that contains $x$.

Taking it account that for every $n\in\mathbb N$ the family $\mathfrak C_n$ is linearly ordered, we conclude that for every $x\in \omega$ the family $\{\mathcal P(x):\mathcal P\in\mathfrak C_n\}$ is linearly ordered and consists of sets of cardinality $\le n$. Consequently, the union $$\mathcal P_n(x):=\bigcup_{\mathcal P\in\mathfrak C_n}\mathcal P(x)$$ is a set of cardinality $\le n$. It can be shown that $\mathcal P_n:=\{\mathcal P_n(x):x\in\omega\}$ is a partition of $\omega$ such that every partition $\mathcal P\in\mathfrak C_n$ refines $\mathcal P_n$.

Since the family $\mathfrak P=\{\mathcal P_n\}_{n\in\mathbb N}$ is countable, it possesses a countable $\mathfrak P$-discrete set, which remains $\mathfrak C$-discrete.

In light of this solution let us ask another

Question. Is any linearly ordered family of finitary partitions of $\omega$ at most countable?