RO-TST 2014

Let $n$ be an integer greater than $1$ and let $S$ be a finite set containing more than $n+1$ elements.Consider the collection of all sets $A$ of subsets of $S$ satisfying the following two conditions :(a) Each member of $A$ contains at least $n$ elements of $S$ .(b) Each element of $S$ is contained in at least $n$ members of $A$ .

Determine $\max_A \min_B |B|$ , as $B$ runs through all subsets of $A$ whose members cover $S$ , and $A$ runs through the above collection.