Cross -intersecting families on set systems

Let L = { l 1 , l 2 , … , l s } be a set of s positive integers. Suppose that A = { A 1 , A 2 , … , A m } and B = { B 1 , B 2 , … , B m } are two collections of subsets of [ n ] = { 1 , 2 , … , n } such that | A i ∩ B j | ∈ L whenever i ≠ j . If the set systems satisfy one of the following conditions: A i ∩ B i | ∈ L implies A i = B i A i ∩ B j | ≤ | A j ∩ B j | with equality possible only when | A i ∩ B i | > | A j ∩ B j | for i ≠ j , then we bound m as m ≤ n − 1 s + n − 1 s − 1 + ⋯ + n − 1 0 . This result extends Snevily’s theorem to cross L -intersecting two families.