= Downward closure of a set family
{title2=$\downarrow\mathcal A$}
= Downward closures of set families
{synonym}
For a <set family> $\mathcal A\subseteq\mathcal P([n])$, its downward closure is $\downarrow\mathcal A=\{S:\text{some }A\in\mathcal A\text{ has }S\subseteq A\}$. It is the smallest <down-set> containing $\mathcal A$, just as the <upward closure of a set family> is the smallest <up-set> containing it. The two closures allow <Harris' inequality> to control sizes of a <cross-Sperner family> pair.
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