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 $m\ge \lceil {\log }_{2}n/\lambda \rceil$.