= Solution
The <extremal number> $\operatorname{ex}(n,H)$ is the maximum number of <edges> in an $n$-vertex <graph> containing no <subgraph> <isomorphic graphs>[isomorphic] to $H$.
Let $G$ be <triangle-free graph>[triangle-free] with $e$ edges. For every edge $uv$, the <neighbourhoods of a vertex>[neighbourhoods] of $u$ and $v$ are disjoint apart from their endpoints, so
$d(u)+d(v)\leq n$. Summing over edges and applying <Cauchy-Schwarz inequality> gives
$$
\sum_vd(v)^2=\sum_{uv\in E}(d(u)+d(v))\leq ne,
$$
while
$$
\sum_vd(v)^2\geq\frac{(\sum_vd(v))^2}{n}
=\frac{4e^2}{n}.
$$
\b[Thus $e\leq n^2/4$, proving the required <Mantel theorem> bound.]
Back to article page
