A new upper bound for cancellative pairs (Q1753087)

Summary: A pair \((\mathcal{A},\mathcal{B})\) of families of subsets of an \(n\)-element set is called cancellative if whenever \(A,A'\in\mathcal{A}\) and \(B\in\mathcal{B}\) satisfy \(A\cup B=A'\cup B\), then \(A=A'\), and whenever \(A\in\mathcal{A}\) and \(B,B'\in\mathcal{B}\) satisfy \(A\cup B=A\cup B'\), then \(B=B'\). It is known that there exist cancellative pairs with \(|\mathcal{A}||\mathcal{B}|\) about \(2.25^n\), whereas the best known upper bound on this quantity is \(2.3264^n\). In this paper we improve this upper bound to \(2.2682^n\). Our result also improves the best known upper bound for Simonyi's sandglass conjecture for set systems.