(i) Define the Turán graph $T_{r}(n)$. State and prove Turán's theorem.

(ii) For each value of $n$ and $r$ with $n>r$, exhibit a graph $G$ on $n$ vertices that has fewer edges than $T_{r-1}(n)$ and yet is maximal $K_{r}$-free (meaning that $G$ contains no $K_{r}$ but the addition of any edge to $G$ produces a $K_{r}$ ). In the case $r=3$, determine the smallest number of edges that such a $G$ can have.