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Is there a common name for a family $\mathscr{F}$ which satisfies the following condition?

For any infinite $X\subseteq\mathscr{F}$ there exists a finite $A\subseteq X$ such that $\bigcap A$ is finite.

Is there a reference on this type of families?

Note that every almost disjoint family satisfies the above condition.


Motivation:

I call a family $\mathscr{F}$ strongly almost disjoint if there is an $n\in\omega$ such that $|A\cap B|<n$ for any two distinct elements $A$, $B$ of $\mathscr{F}$.

Using ideas in Specker's and Halbeisen's papers, I proved in $\mathsf{ZF}$ that for any infinite set $M$ and any strongly almost disjoint $\mathscr{F}\subseteq\mathscr{P}(M)$, $|\mathscr{F}|<|\mathscr{P}(M)|$.

Recently, I found that the proof also works for any family $\mathscr{F}$ which satisfies the following condition:

There is an $n\in\omega$ such that for any infinite $X\subseteq\mathscr{F}$ there exists a finite $A\subseteq X$ for which $|\bigcap A|<n$.

So I ask whether there is a known name for such a family if we replace $``|\bigcap A|<n"$ by $``\bigcap A$ is finite$"$.

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    $\begingroup$ I don't know, but it seems to me that your property is somewhat easier to understand if we regard the family of sets as a bipartite graph, so it may be that it has been considered and named in the literature on infinite graphs. Doesn't your condition just say that the bipartite graph does not contain a certain countable bipartite graph? So if that graph happens to have a common name, say $H_\omega$, then you could call your things $H_\omega$-free bipartite graphs. $\endgroup$ – bof May 9 at 8:45
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    $\begingroup$ If nothing else comes up, then "weakly/quasi/almost almost disjoint" could work. $\endgroup$ – Asaf Karagila May 9 at 9:48
  • $\begingroup$ Asaf, bof, Thank you for your comments. $\endgroup$ – Guozhen Shen May 9 at 10:16

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