On the Maximum of the Sum of the Sizes of Non-trivial Cross-Intersecting Families

Let n ≥ 2k ≥ 4 be integers, [n] ()k the collection of k -subsets of [n] = {1, … , n} . Two families ℱ, 𝒢⊂[n] ()k are said to be cross-intersecting if F ∩ G ∅ for all F ∈ℱ and G ∈𝒢 . A family is called non-trivial if the intersection of all its members is empty. The best possible bound |ℱ| + |𝒢| ≤n ()k - 2 n - k ()k + n - 2k ()k + 2 is established under the assumption that ℱ and 𝒢 are non-trivial and cross-intersecting. For the proof a strengthened version of the so-called shifting technique is introduced. The most general result is Theorem 4.1 .