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

摘要

LetSbe a finite set, and fixK>2. LetFbe a family of subsets ofSwith the property that wheneverA1,...,Akare sets inF, not necessarily distinct, andA1⋂ ⋯ ⋂Ak= ∅, thenA1⋃ ⋯ ⋃Ak=S. We prove here that the maximum size of such a family is 2S−1+ 1. If we require that the setsA1,...,Akbe distinct, then the maximum size ofFis again 2S−1+ 1, provided that