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

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 $${\cal Q}$$.)

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?

• Thanks for the beautiful answer - and if your question in turn has a negative answer, we might see yet another cardinal characteristic :-) . If an answer to your question is not found over the next few days, it could be worthwhile that you ask this as a separate MO question Feb 19, 2020 at 22:41