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If $l>k$ then the required condition always holds, so we assume that $1\le l\le k$. Let $n=|S|$. Then there are ${n\choose l}$ subsets of size $l$ of $S$, and each of the given $r$ subsets contains ${k\choose l}$ of subsets of size $l$. Therefore $r\cdot {k\choose l}\le {n\choose l}$, which provides a lower bound for $n$.