Let $G$ be a finite, simple, loopless graph with $|V(G)|=n$. We define its *edge density* as $$ed(G) := \frac{|E(G)|}{n \choose 2}.$$


Moreover we set $$d\_n := \text{max}\big\{ed(G): G \text{ is a triangle-free graph with } V(G) = \{1,\ldots, n\} \big\}.$$


Does $\lim\_{n\to\infty}d\_n$ exist and if yes, what is its value? If not, what is $\lim \text{sup}\_{n\to\infty}d\_n$? (I would expect this value to be around $2/3$ but I really don't have any idea.)



Edge density of triangle-free graphs
[Turan's theorem](http://en.wikipedia.org/wiki/Tur%C3%A1n%27s_theorem) says that this limit exists and is equal to $\frac{1}{2}$. The graphs that achieve this are the complete bipartite graphs. Moreover, the theorem gives the answer for the more general question of $K\_{r+1}$-free graphs, where maximum edge density comes from complete $r$-partite graphs.


