 Solution

Put $m=\lceil {\log }_{2}n\rceil$ and assign distinct complete binary sequences in $\{0,1{\}}^m$ to the $n$ points. For each coordinate $i$, let $A_i$ be the points with bit zero and $B_i$ those with bit one. Distinct sequences disagree in some coordinate, so these pairs form a separating family of disjoint set pairs.

Every point lies in exactly one side of every pair, hence the total incidence is $mn$, as allowed when $\lambda =1$. The construction reaches the bound:The reference to “(i)” in the original PDF's part (b) refers to part (a); the local TeX additionally duplicates that item.