New Answer: It indeed follows from vertex-multiplication. If you replace each edge with a stable set of size $k$, then $\alpha(G)/n$ will not change, while $n^r$ and the number of edges both grow by a $k^r$ factor. This shows that $\lim_{n \to \infty}\frac{RT(n,L^{(r)},\varepsilon n)}{n^r}$ has a limit by a Fekete lemma type argument.

Old Answer: Many of these functions are not known to have a limit. In this case when someone proves a bound on them, in fact they show a bound for the liminf or limsup. Unfortunately, in many papers they refer to it as lim, which also bothers me a lot. Incidentally, I've just asked the same thing two weeks ago (about the uniform Turan density function) from Danial Kral'.

New Answer: It indeed follows from vertex-multiplication. If you replace each vertex with a stable set of size $k$, then $\alpha(G)/n$ will not change, while $n^r$ and the number of edges both grow by a $k^r$ factor. This shows that $\lim_{n \to \infty}\frac{RT(n,L^{(r)},\varepsilon n)}{n^r}$ has a limit by a Fekete lemma type argument.

Old Answer: Many of these functions are not known to have a limit. In this case when someone proves a bound on them, in fact they show a bound for the liminf or limsup. Unfortunately, in many papers they refer to it as lim, which also bothers me a lot. Incidentally, I've just asked the same thing two weeks ago (about the uniform Turan density function) from Danial Kral'.

Many of these functions are not known to have a limit. In this case when someone proves a bound on them, in fact they show a bound for the liminf or limsup. Unfortunately, in many papers they refer to it as lim, which also bothers me a lot. Incidentally, I've just asked the same thing two weeks ago (about the uniform Turan density function) from Danial Kral'.

New Answer: It indeed follows from vertex-multiplication. If you replace each edge with a stable set of size $k$, then $\alpha(G)/n$ will not change, while $n^r$ and the number of edges both grow by a $k^r$ factor. This shows that $\lim_{n \to \infty}\frac{RT(n,L^{(r)},\varepsilon n)}{n^r}$ has a limit by a Fekete lemma type argument.

Old Answer: Many of these functions are not known to have a limit. In this case when someone proves a bound on them, in fact they show a bound for the liminf or limsup. Unfortunately, in many papers they refer to it as lim, which also bothers me a lot. Incidentally, I've just asked the same thing two weeks ago (about the uniform Turan density function) from Danial Kral'.