Using Turan's theorem to calculate a bound for triangle free graphs.

Question

Using $$ \alpha(G) \ge \frac{n^2}{2\mathbb{E}(G) + n} $$ prove that any graph on n vertices with no triangles has at most $\frac{n^2}{4}$ edges.

$\alpha(G)$ is the maximum size of independent vertices in G.

Answer 1

Let $G^{'}$ be a triangle free graph, and that $G$ is its complement.

From this we get that $\alpha(G) \le 2$ (Why?)

Finally we get $$\mathbb{E}(G) \ge \frac{n^2}{4} - \frac{n}{2}$$

After one more step, the proof is finished for $G^{'}$. (How?)