Separating family of disjoint set pairs

A family $(A_i,B_i{)}_{i=1}^m$ of pairs of disjoint subsets of a finite set separates its points if each two distinct points lie on opposite sides of at least one pair. Empty sides are allowed by this definition. Let $d(x)$ count pairs containing $x$. Associate to $x$ the subcube of $\{0,1{\}}^m$ fixing coordinate $i$ to zero on $A_i$ and one on $B_i$. Separation makes these subcubes disjoint and yields the disjoint subcube packing inequality.

Pairwise disjoint subcubes of $\{0,1{\}}^m$ fixing $d_1,\dots ,d_n$ coordinates satisfy ${\sum }_{x=1}^n{2}^{-d_x}\le 1$. By Jensen inequality, their average codimension is at least ${\log }_{2}n$. Consequently a separating family of disjoint set pairs of total incidence at most $\lambda mn$, with $\lambda >0$, has $m\ge \lceil {\log }_{2}n/\lambda \rceil$.