I know that it must be a simple consequence of the Kővári–Sós–Turán (and Erdős–Stone) theorem, but I am struggling to formulate a proof: Let $H$ be a fixed-size $r$-chromatic graph. Then there exists $\varepsilon = \varepsilon(H)$ s.t. if $G$ is an $n$-vertex graph that contains $\geq n^{r-\varepsilon}$ copies of $K\_r$, it must also contain a (not necessarily induced) copy of $H$.


Thanks in advance.



Extremal graph theory - many copies of $K_r$ imply a copy of $r$-chromatic $H$
This follows from Proposition 2.1 of the paper [Many $T$ copies in $H$-free graphs](https://arxiv.org/abs/1409.4192) by Alon and Shikhelman.


**Theorem** (Alon and Shikelman)
Let $T$ be a fixed graph with $t$ vertices. Then $ex(n,T,H)=\Omega(n^t)$ if and only if $H$ is not a subgraph of a blow-up of $T$. Otherwise, $ex(n,T,H) \le n^{t-\varepsilon(T,H)}$ for some $\varepsilon(T,H) > 0$.


Here, $ex(n,T,H)$ is the maximum number of copies of $T$ in an $n$-vertex $H$-free graph.


To derive your result, we apply Alon-Shikelman with $T=K\_r$ and $H$ a fixed $r$-chromatic graph. Thus, $H$ is a subgraph of a blow-up of $K\_r$. So, we may choose $\varepsilon(H)$ to be any positive constant less than $\varepsilon(K\_r, H)$.


