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Combinatorics Difficulty 7.0 National Olympiad, round 2 Prove it Hungary

We call a system of non-empty sets $H$ entwined, if for every disjoint pair of sets $A$ and $B$ in $H$ there exists $b\in B$ such that $A\cup\{b\}$ is in $H$ or there exists $a\in A$ such that $B\cup\{a\}$ is in $H$ .

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