  In this very short paper, we point out that the average overlap density of a
union-closed family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ may be as
small as $\Theta((\log \log |\mathcal{F}|)/(\log |\mathcal{F}|))$, for
infinitely many positive integers $n$.
