An Extremal Problem for two Families of Sets

Let r and s be positive integers and { A 1 A 2 , ... , A m }, { B 1 , ... , B m } be two families of sets with | A i |= r | B i |= s such that A i ∩ B i =∅ and A i ∩ B i ≠∅ for every i = 1, ..., m and i m ≤ ( r + s s ) . This result generalizes a theorem of Bollobas, The problem was raised by Pin. The proof uses linear algebra and symmetric tensor products.