Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 30J / d / Solution

For each $d$-element coordinate set $J\subseteq \{1,\dots ,p\}$, let ${\mathcal{G}}_J$ contain the lower orthants whose finite thresholds occur precisely in the coordinates in $J$. Ignoring the remaining coordinates identifies ${\mathcal{G}}_J$ with the lower-orthant class in ${\mathbb{R}}^d$, so part (a) gives $\mathrm{VC}({\mathcal{G}}_J)=d$. Moreover,is a union of $r=\left(\genfrac{}{}{0ex}{}{p}{d}\right)$ such classes. Part (c) and $\left(\genfrac{}{}{0ex}{}{p}{d}\right)\le p^d$ give