Some best possible inequalities concerning cross-intersecting families

Abstract

Let A be a non-empty family of a-subsets of an n-element set and B a non-empty family of b-subsets satisfying A ∩ B ≠ ⊘ for all A ∈ A , B ∈ B . Suppose that n⩾ a+ b, b⩾ a. It is proved that in this case | A |+| B |⩽( n b )− n−a b ) holds. Various extensions of this result are proved. Two new proofs of the Hilton-Milner theorem on non-trivial intersection families are given as well.