Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 17F / a / Solution

The extremal number $\mathrm{ex}(n,H)$ is the maximum number of edges in an $n$-vertex graph containing no subgraph isomorphic to $H$.

Let $G$ be triangle-free with $e$ edges. For every edge $uv$, the neighbourhoods of $u$ and $v$ are disjoint apart from their endpoints, so
$d(u)+d(v)\le n$. Summing over edges and applying Cauchy-Schwarz inequality giveswhileThus $e\le n^2/4$, proving the required Mantel theorem bound.

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