Solution
= Solution
An even cycle is bipartite and has <chromatic number> two. An odd cycle has <chromatic number> three. The extremal-density theorem therefore gives
$$
\boxed{\lim_{n\to\infty}\frac{\operatorname{ex}(n;C_k)}{\binom n2}=\begin{cases}0,&k\text{ even},\\1/2,&k\text{ odd}.\end{cases}}
$$
For odd cycles the complete balanced <bipartite graph> also exhibits the <matching> asymptotic lower density one half.