A note on the maximum product-size of non-trivial cross t-intersecting families.

Two families A and B are called cross t -intersecting if | A ∩ B | ≥ t for all A ∈ A , B ∈ B. Let n , k , t be positive integers and A , B ⊆ ( n k ). Pyber proved that if A and B are cross 1-intersecting, then | A | | B | ≤ ( n − 1 k − 1 ) 2 for all n ≥ 2 k , and equality holds if and only if A = B = { A ∈ ( [ n ] k ) : x ∈ A } for some x ∈ [ n ] whenever n > 2 k. Frankl and Kupavskii sharpened Pyber's inequality to | A | | B | ≤ (( n − 1 k − 1 ) + ( n − i k − i + 1 )) (( n − 1 k − 1 ) − ( n − i k − 1 )) under the assumption that | B | ≥ ( n − 1 k − 1 ) + ( n − i k − i + 1 ). In this paper, firstly, we show that the families that attain this bound are unique for n ≥ 2 k + 1. Secondly, combining the result of Frankl and Kupavskii, we show that if A ⊆ ( [ n ] k ) and B ⊆ ( [ n ] k ) are cross 1-intersecting with ∩ B ∈ B B = ∅ , then | A | | B | ≤ (( n − 1 k − 1 ) − ( n − k − 1 k − 1 )) (( n − 1 k − 1 ) + 1). Lastly, we generalize the above result to cross t -intersecting families for all large n. [ABSTRACT FROM AUTHOR]