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In my homework, I'm asked to prove the following:

Let $b_n(r,\epsilon)$ be the largest integer $b$ such that any graph with $\left(1-\frac{1}{r} +\epsilon\right)\frac{n^2}{2}$ edges contains a $b$-blowup of $K_{r+1}$ (meaning, that it contains a complete $r+1$-partite graph, with $b$ vertices in each partition).

I need to show that $b_n(1,\epsilon)=\Theta\left(\frac{\log n}{\log(1/\epsilon)}\right)$ and that $b_n(2,\epsilon)=O\left(\frac{\log n}{\log(1/\epsilon)}\right)$.

First, I'm not sure how to show either the lower bound, or the upper bound. For the first part, I know that $ex(n; K_{b,b}) \geq c\binom{n}{2}^{1-\frac{1}{t+1}}$ and $ex(n; K_{b,b}) \leq cb^{\frac{1}{b}}n^{2-\frac{1}{b}}$ and I think that for one of the bounds I need to combine one of those inequalities, with the fact that for $\epsilon\binom{n}{2}$ there is a bipartite graph.

I'll appreciate any light you can shed on the subject.

Thanks in advance.

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