New bounds on Simonyi’s conjecture.

We say that a pair ( A , B ) is a recovering pair if A and B are set systems on an n -element ground set, such that for every A , A ′ ∈ A and B , B ′ ∈ B we have ( A ∖ B = A ′ ∖ B ′ implies A = A ′ ) and symmetrically ( B ∖ A = B ′ ∖ A ′ implies B = B ′ ). G. Simonyi conjectured that if ( A , B ) is a recovering pair, then | A | | B | ≤ 2 n . For the quantity | A | | B | the best known upper bound is 2 . 326 4 n due to Holzman and Körner. In this paper we improve this upper bound to 2 . 281 4 n . [ABSTRACT FROM AUTHOR]