Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 17I / a / Solution

WriteLet $G$ be an $H$-free graph on $n$ vertices with $\mathrm{ex}(n,H)$ edges, and choose a uniformly random set of $m<n$ vertices. Its induced subgraph is still $H$-free, and each edge survives with probability $\left(\genfrac{}{}{0ex}{}{m}{2}\right)/\left(\genfrac{}{}{0ex}{}{n}{2}\right)$. Henceso $d_m(H)\ge d_n(H)$. Thus the normalized extremal number is nonincreasing and bounded below by zero, proving that