Cancellative pairs of families of sets

A pair (A, B) of families of subsets of an n-element set X is cancellative if, for all A, A′ ϵ A and B, B′ ϵ B, the following conditions hold: A⧹B = A′⧹B ⇒ A = A′ and B⧹A = B′⧹A ⇒ B = B′. We prove that every such pair satisfies AB\̌gd , where τ = 2.3264. This is related to a conjecture of Erdõ and Katona on cancellative families and to a conjecture of Simonyi on recovering pairs. For the latter, our result gives the best known upper bound.