On Cross-intersecting Families of Sets

A family [inline-graphic not available: see fulltext] of ℓ-element subsets and a family [inline-graphic not available: see fulltext] of k-element subsets of an n-element set are cross-intersecting if every set from [inline-graphic not available: see fulltext] has a nonempty intersection with every set from [inline-graphic not available: see fulltext]. We compare two previously established inequalities each related to the maximization of the product [inline-graphic not available: see fulltext], and give a new and short proof for one of them. We also determine the maximum of [inline-graphic not available: see fulltext] for arbitrary positive weights ωℓ,ωk.