On cancellative pairs of families of subsets

Abstract: A pair $(\mathcal{A}, \mathcal{B})$ of families of subsets of $[n]$ is cancellative if whenever $A, A&#39; \in \mathcal{A}, B \in \mathcal{B}$ satisfy $A \cup B=A&#39; \cup B$, then $A=A&#39;$, and whenever $A \in \mathcal{A}, B, B&#39; \in \mathcal{B}$ satisfy $A \cup B=A \cup B&#39;$, then $B=B&#39;$. We show that for every cancellative pair $(\mathcal{A}, \mathcal{B})$, the inequality $|\mathcal{A}||\mathcal{B}| \le 2.25<sup>n$ holds, matching Tolhuizen's $(2.25-o(1))<sup>n$ lower bound construction.