On cancellative pairs of families of subsets

A pair $(\mathcal{A},\mathcal{B})$ of families of subsets of $[n]$ is cancellative if whenever $A,A^{\mathrm{\prime }}\in \mathcal{A},B\in \mathcal{B}$ satisfy $A\cup B=A^{\mathrm{\prime }}\cup B$, then $A=A^{\mathrm{\prime }}$, and whenever $A\in \mathcal{A},B,B^{\mathrm{\prime }}\in \mathcal{B}$ satisfy $A\cup B=A\cup B^{\mathrm{\prime }}$, then $B=B^{\mathrm{\prime }}$. We show that for every cancellative pair $(\mathcal{A},\mathcal{B})$, the inequality $\mathrm{\mid }\mathcal{A}\mathrm{\mid }\mathrm{\mid }\mathcal{B}\mathrm{\mid }\le {2.25}^n$ holds, matching Tolhuizen’s $(2.25-o(1){)}^n$ lower bound construction.