Some Results on Systems of Finite Sets That Satisfy a Certain Intersection Condition

Let S be a finite set, and fix K>2. Let F be a family of subsets of S with the property that whenever A₁,...,A_k are sets in F, not necessarily distinct, and A₁ ? ? ? A_k = ?, then A₁ ? ? ? A_k = S. We prove here that the maximum size of such a family is 2^|S|?1 + 1. If we require that the sets A₁,...,A_k be distinct, then the maximum size of F is again 2^|S|?1 + 1, provided that |S| ≥ log₂(K?2)+3.