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

Let \(n \ge 2k \ge 4\) be integers, \({[n]\atopwithdelims ()k}\) the collection of k-subsets of \([n] = \{1, \ldots , n\}\). Two families \({\mathcal {F}}, {\mathcal {G}} \subset {[n]\atopwithdelims ()k}\) are said to be cross-intersecting if \(F \cap G \ne \emptyset \) for all \(F \in {\mathcal {F}}\) and \(G \in {\mathcal {G}}\). A family is called non-trivial if the intersection of all its members is empty. The best possible bound \(|{\mathcal {F}}| + |{\mathcal {G}}| \le {n \atopwithdelims ()k} - 2 {n - k\atopwithdelims ()k} + {n - 2k \atopwithdelims ()k} + 2\) is established under the assumption that \({\mathcal {F}}\) and \({\mathcal {G}}\) 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.