Let $\mathcal{A} \subset 2^{\{1, 2, \dots, n\}}$ be a unionclosed 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/upwardclosed family.
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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 UnionClosed Sets Conjecture" by H. Bruhn and O. Schaudt which may have some pointers to something useful. $\endgroup$– Louis DFeb 11, 2022 at 2:14
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