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

Abstract

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