Define
$$RT(n,K\_l,f(n))=ex\_l(n,f(n))=\max\_G\{e(G): K\_l \not\subset G, v(G)=n, \alpha(G)\leq f(n)\}$$
and the Ramsey-Turán density function $f\_l:(0,1] \to \mathbb{R}$ as
$$f\_l(\alpha)=\lim\_{n\to \infty}\frac{ex\_l(n,\alpha n)}{{n\choose 2}}.$$


By Turán Theorem we have that $f\_l(\alpha)=1-\frac{1}{l-1}$ for every $\alpha \geq \frac{1}{l-1}$.


How to prove that this limit exists for every $\alpha \in (0,1]$?


I was trying to prove that the sequence $(ex\_l(n,\alpha n)/{n\choose 2})\_n$ is monotone, but I could only prove that
$$ex\_l(n,s)\leq\frac{n \ ex\_l(n-1,s)}{n-2},$$
which is not enough since $ex\_l(n-1,\alpha n) \geq ex\_l(n-1,\alpha(n-1)).$


I also thought of using that $\alpha(G) \leq \alpha n$ and $\alpha(G)\leq \alpha(n-1)$ only give different restrictions once every $\approx 1/\alpha$ values of $n$ in a row. But it would also be necessary to show that every jump is in the same direction, which I couldn't do.


**Edit 1:** In [Simonovits and Sós](https://www.semanticscholar.org/paper/Ramsey-Tur%C3%A1n-theory-Simonovits-S%C3%B3s/b6acab1b394d5096f2cd7221022553cc2cad2f7d) survey on Ramsey-Turán theory, when defining $RT(n,L^{(r)},s)$ for $r$-uniform hypergraphs, they say that the limit
$$\lim\_{n \to \infty}\frac{RT(n,L^{(r)},\varepsilon n)}{n^r}$$
exists and follows relatively easily from vertex-multiplication. I don't quite know how vertex-multiplication works but maybe it helps.



Ramsey-Turán density function is well defined
I answered the question in detail for $RT(n, K\_l, \alpha n)$ [here](https://math.stackexchange.com/a/4282372/383078) on Math StackExchange.


