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 ${\cal Q}$.)

1 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?

  • $\begingroup$ 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 $\endgroup$ Feb 19, 2020 at 22:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.