HMMT 2002 E4

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?