# Turán's Theorem: clique-free graph and minimum independent set size

I find two different formulations of Turán's theorem and I don't see why they are equivalent.

One is:

(1) Let $$G$$ be any graph with $$n$$ vertices, such that $$G$$ is $$K_{r+1}$$ -free. Then the number of edges in $$G$$ is at most

$$\frac{r-1}{r}.\frac{n^2}{2}$$

(for example from Wikipedia)

The other one is:

(2) If $$G$$ is a graph with $$n$$ vertices and $$e$$ edges then there is an independent set of size at least $$\frac{n}{\frac{2e}{n} + 1}$$

(for example from this document)

Let $G'$ be the complement graph for $G$, $r$ be the maximal size of a clique of the graph $G$ (which is equal to the maximal size of an independent set of the graph $G'$), and $r'$ be the maximal size of a clique of the graph $G'$ (which is equal to the maximal size of an independent set of the graph $G$).
(1) $\Rightarrow$ (2). The graph $G'$ is $K_{r'+1}$ free and therefore $\frac{n(n-1)}2-e(G)=e(G')\le\frac{r'-1}{r'}\frac{n^2}{2}$. Simplifying the last inequality we obtain (2).
(2) $\Rightarrow$ (1). In this case $r\ge \frac{n^2}{2e(G') + n}= \frac{n^2}{n(n-1)-2e(G)+ n}$. Simplifying the last inequality we obtain (1).