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For any finite set $S$ and every partition $S_1, \dots, S_n$ of $S$, let $P(S_1, \dots, S_n)$ be the family consisting of all possible unions of $S_1, \dots, S_n$. Clearly, $P(S_1, \dots, S_n)$ is a union-closed family and all elements of $S$ are abundant (present in at least half the sets of $P(S_1, \dots, S_n)$).

Do all union-closed families such that all elements are abundant come from partitions?

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  • $\begingroup$ No, the family $\{S\}$ has this property. $\endgroup$
    – Tony Huynh
    Commented Feb 7, 2018 at 15:33
  • $\begingroup$ I meant, apart from this trivial one. I will edit the question accordingly. Thanks! $\endgroup$
    – Sisyphus
    Commented Feb 7, 2018 at 15:41
  • $\begingroup$ I think all elements will be abundant when the basis sets partition the global set S. Want to know if there are any other situations where all elements are abundant. $\endgroup$
    – Sisyphus
    Commented Feb 7, 2018 at 16:15
  • $\begingroup$ I edited the question to reflect what I think you mean. $\endgroup$
    – Tony Huynh
    Commented Feb 7, 2018 at 16:57

1 Answer 1

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no, consider all subsets $A$ of a given set $S$ such that $|A|\geqslant 5$

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