Define the Turán graph $T_{r}(n)$. State and prove Turán's theorem. Hence, or otherwise, find $\operatorname{ex}\left(K_{3} ; n\right)$.

Let $G$ be a bipartite graph with $n$ vertices in each class. Let $k$ be an integer, $1 \leqslant k \leqslant n$, and assume $e(G)>(k-1) n$. Show that $G$ contains a set of $k$ independent edges.

[Hint: Suppose $G$ contains a set $D$ of a independent edges but no set of a $+1$ independent edges. Let $U$ be the set of vertices of the edges in $D$ and let $F$ be the set of edges in $G$ with precisely one vertex in $U$; consider $|F| .]$

Hence, or otherwise, show that if $H$ is a triangle-free tripartite graph with $n$ vertices in each class then $e(H) \leqslant 2 n^{2}$.