Separating set system ($\forall i\ne j\exists A:i\in A,j\notin A$)

A set family on a finite set separates ordered pairs if each ordered pair of distinct points has a member containing the first point and omitting the second. If its members are $A_1,\dots ,A_m$, assign each point the characteristic vector of its incidences, or equivalently $T_i=\{a:i\in A_a\}$. Separation in both directions says precisely that the $T_i$ form an antichain in the Boolean lattice on $m$ coordinates. Thus the Sperner theorem gives $n\le \left(\genfrac{}{}{0ex}{}{m}{\lfloor m/2\rfloor }\right)$. Conversely, any antichain of $n$ distinct subsets of $[m]$ supplies such a set family by transposing its incidence table.