Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 109 / 2 / c

For $n=4$, the lower bound is four pairs. On $X=\{1,2,3,4\}$ useThe first pair separates points in different halves, and the next two separate points within each half. The four pairs are distinct and have total incidence $4+2+2+0=8=(1/2)\cdot 4\cdot 4$. They are therefore a valid separating family of disjoint set pairs attaining the bound.

For $n=8$, label points by triples $(u,v,w)\in \{0,1{\}}^3$. The following six pairs attain the bound of six. Coordinates not mentioned in a row are unrestricted:If two triples first differ in $u$, row 1 separates them. If they agree in $u$ and differ in $v$, row 2 or 3 does. If they agree in both and differ in $w$, row 4, 5, or 6 does. The total incidence is $24=(1/2)\cdot 6\cdot 8$, and each point belongs to exactly three pairs. ThusThe empty fourth pair matters for $n=4$. The original PDF requires disjoint subsets but does not require either side to be nonempty, so the construction is permitted. Under the additional convention that both sides of every pair must be nonempty, the $n=4$ answer would be no: four pairs of total incidence at most eight would each have two singleton sides and could separate at most four of the six unordered pairs of points. The $n=8$ construction already has both sides nonempty.