HMMT 二月 2002 · 几何 · 第 3 题

(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 ∈ C, R ∈ 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 · 2 = 6 committees. This minimum is attainable - just take four professors and let any two professors form a committee.