Let $\mathcal{A} \subset 2^{\{1, 2, \dots, n\}}$ be a union-closed family of sets (i.e., for any $A, B \in \mathcal{A}$, also $A \cup B \in \mathcal{A}$). In this paper it is established that $$ \sum_{A \in \mathcal{A}} |A| \geq \frac{1}{2} |\mathcal{A}| \log_2(|\mathcal{A}|). $$ Is there a fundamentally different proof of this result to the one given in the cited paper? In particular, I am looking for a proof which does not require the concept of a filter/upward-closed family.

  • 1
    $\begingroup$ This probably doesn't answer your question, but there is a discussion of Reimer's proof in Section 6.4 of "The Journey of the Union-Closed Sets Conjecture" by H. Bruhn and O. Schaudt which may have some pointers to something useful. $\endgroup$
    – Louis D
    Feb 11, 2022 at 2:14


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