What's the largest constant $c >1$ such that all triangle free graphs with chromatic number $n$ has atleast $\Theta(c^n) $ vertices? 



Size of triangle free graph with chromatic number $n$
It follows from the result in [A note on Ramsey numbers](https://www.sciencedirect.com/science/article/pii/0097316580900308) by M. Ajtai, J. Komlós, E. Szemerédi that a triangle free graph on $s$ vertices has chromatic number at most $\Theta(\sqrt\frac{s}{\log s})$.


Moreover, in [The Ramsey Number $R(3, t)$ has Order of Magnitude $t^2/\log t$](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.88.239&rep=rep1&type=pdf), Kim proves that this is optimal by exhibiting triangle free graphs on $s$ vertices with chromatic number equal to that bound. This means that if a triangle free graph has chromatic number $n$,as in the question, its number of vertices can be as low as $\Theta(n^2\log n)$ which is not exponential in $n$, as the question suggests.


