4. We call a set of professors and committees on which they serve a university if
(1) given two distinct professors there is one and only one committee on which they both serve,
(2) given any committee, $C$, and any professor, $P$, not on that committee, there is exactly one committee on which $P$ serves and no professors on committee $C$ serve, and
(3) there are at least two professors on each committee; there are at least two committees.
What is the smallest number of committees a university can have?
Solution: Let $C$ be any committee. Then there exists a professor $P$ not on $C$ (or else there would be no other committees). By axiom 2, $P$ serves on a committee $D$ having no common members with $C$. Each of these committees has at least two members, and for each $Q \in C, R \in D$, there exists (by axiom 1) a committee containing $Q$ and $R$, which (again by axiom 1) has no other common members with $C$ or $D$. Thus we have at least
$2+2 \cdot 2=6$ committees. This minimum is attainable - just take four professors and let any two professors form a committee.