Let C be a finite collection of finite sets. Further let C be closed under union. Is it always the case that there is an element x which is in at least half the members of C?

The answer to this question is no. If, however, the union of C is required to be nonempty, then the answer is unknown to me, and I suspect to many others. (Look up union closed sets conjecture.)

Gerhard "Ask Me About System Design" Paseman, 2012.02.26