On Non-Empty Cross-Intersecting Families

Let 2 [ n ] and ( [ n ] i ) be the power set and the collection of all i -subsets of {1, 2, …, n }, respectively. We call t ( t ≥ 2) families A 1 , A 2 , … , A t ⊆ 2 [ n ] cross-intersecting if A i ∩ A j ≠ ∅ for any A i ∈ A i and A j ∈ A j with i ≠ j . We show that, for n ≥ k + l, l ≥ r ≥ 1, c > 0 and A ⊆ ( [ n ] k ) , ℬ ⊆ ( [ n ] l ) , if A and ℬ are cross-intersecting and ( n − r l − r ) ≤ | ℬ | ≤ ( n − 1 l − 1 ) , then | A | + c | ℬ | ≤ max { ( n k ) − ( n − r k ) + c ( n − r l − r ) , ( n − 1 k − 1 ) + c ( n − 1 l − 1 ) } . This implies a result of Tokushige and the second author (Theorem 3.1) and also yields that, for n ≥ 2 k , if A 1 , A 2 , … , A t ⊆ ( [ n ] k ) are non-empty cross-intersecting, then ∑ i = 1 t | A i | ≤ max { ( n k ) − ( n − k k ) + t − 1 , t ( n − 1 k − 1 ) } , which generalizes the corresponding result of Hilton and Milner for t = 2. Moreover, the extremal families attaining the two upper bounds above are also characterized.