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

Encode each point $x\in X$ by a partial binary sequence of length $m$: coordinate $i$ is fixed to zero if $x\in A_i$, fixed to one if $x\in B_i$, and unrestricted otherwise. Disjointness makes these prescriptions consistent. Let $d(x)$ be the number of fixed coordinates, equivalently the number of pairs containing $x$, and let $S_x\subseteq \{0,1{\}}^m$ be the set of complete sequences consistent with them. Then $|S_x|=2^{m-d(x)}$.

The separating family of disjoint set pairs ensures that $S_x$ and $S_y$ are disjoint whenever $x\ne y$: one coordinate prescribes opposite bits. Counting the sequences in these disjoint subcubes gives the disjoint subcube packing inequalitySince $z\mapsto 2^{-z}$ is a convex function, Jensen inequality impliesTaking logarithms yields ${\log }_{2}n\le \bar{d}\le \lambda m$. Therefore, for $\lambda >0$,The PDF omits the qualification $\lambda >0$. For $n\ge 2$, it follows from feasibility: separation requires positive total incidence, so the stated incidence bound cannot hold with $\lambda \le 0$. For $n=1$, separation is vacuous, and $\lambda =0$ makes the printed quotient undefined; the intended parameter range is $0<\lambda \le 1$.