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I'm following the proof of Turan's theorem on $\text{ex}(n,K^r)$ in Diestel's Graph Theory book (click to see the page) and something bothers me:

Since $G$ is edge-maximal without a $K^r$ subgraph, $G$ has a subgraph $K=K^{r-1}$. By the induction hypothesis, $G-K$ has at most $t_{r-1}(n-r+1)$ edges [...]

But why would $G-K$ be $K^{r-1}-$free ? Couldn't there be other $K^{r-1}$ disjoint subgraphs in $G$ ?

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up vote 1 down vote accepted

The proof does not claim that $G-K$ is $K^{r-1}$-free, only that it is $K^r$-free. The induction is on $n$, not on $r$, so the assumption is that all smaller $K^r$-free graphs have been classified.

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