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    $\begingroup$ If $\mathcal{A}$ is an uncountable almost disjoint family what can we say about $\{|A\cap B|:A,B\in \mathcal{A},A\neq B\}$? Must it be infinite? $\endgroup$ Commented Oct 8, 2024 at 0:27
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    $\begingroup$ @n901 Yes, it must be infinite. If $|A\cap B|\lt n$ whenever $A,B\in\mathcal A$, $A\ne B$, then each $n$-element subset of $\omega$ is contained in at most one element of $\mathcal A$, so $\mathcal A$ is countable. $\endgroup$ Commented Oct 8, 2024 at 3:56
  • $\begingroup$ @bof I don't see why $\mathcal{A}'$ is maximal. $\endgroup$ Commented Oct 8, 2024 at 19:36
  • $\begingroup$ @bof Thank you! $\endgroup$ Commented Oct 8, 2024 at 21:48