Some Results on the Sandglass Conjecture

Let A, B be subsets of a lattice L. The pair (A, B) is a recovering pair if for every a, a ∈ A, b, b1 ∈ B = a′ ∨ b′ ⇒ a = a′ = a′ ∧ b′ ⇒ b = b′ ecture of Simonyi states that in the case when L =2n, the bound ∣A∣∣-B∣ ≤ = 2n holds for recovering pairs. This conjecture is still open. Ahlswede and Simonyi generalized the conjecture for the product of chains of any length, in the sense that {A11B} achieves its maximum for pairs of a special form, which they called sandglasses. We prove the Ahlswede-Simonyi conjecture for any fixed number of sufficiently long chains, and for the product of three chains of any length.