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Given a set $A$ of $n$ elements and an positive integer $k\le n$, what is the size of the largest Sperner family $\mathcal{F}$ of subsets of $A$ such that $\mathcal{F}$ contains a set $B\subseteq A$ of size $k$?

I wonder whether it is just the family made of all subsets of $A$ of size $k$ (and why) or one can build a larger one.

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