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Set family $\mathcal{F}$ such that for all $A,B,C \in \mathcal{F}$ both $A \cap B \not \subseteq C$ and $C \not \subseteq A \cup B $
This question initially arose out of a question in asymptotic matroid theory. The matroid question has since been answered in a different way, but the extremal set theory question remains unanswered ...
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