Question 12
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 $\mathcal{A}$ of subsets of $S$ satisfying the following two conditions:
(a) Each member of $\mathcal{A}$ contains at least $n$ elements of $S$; and
(b) Each element of $S$ is contained in at least $n$ members of $\mathcal{A}$.
Determine $\max_{\mathcal{A}} \min_{\mathcal{B}} |\mathcal{B}|$, as $\mathcal{B}$ runs through all subsets of $\mathcal{A}$ whose members cover $S$, and $\mathcal{A}$ runs through the above collection.
(a) Each member of $\mathcal{A}$ contains at least $n$ elements of $S$; and
(b) Each element of $S$ is contained in at least $n$ members of $\mathcal{A}$.
Determine $\max_{\mathcal{A}} \min_{\mathcal{B}} |\mathcal{B}|$, as $\mathcal{B}$ runs through all subsets of $\mathcal{A}$ whose members cover $S$, and $\mathcal{A}$ runs through the above collection.