How large can the chromatic number of an $n$-vertex $C\_4$-free graph be? If the maximum degree of the graph $G$ is $\Delta$, is there a bound of the form 
$\chi(G) \leq O(\Delta/\log(\Delta))$ as in the case of triangles? What happens if $e(G)$ is close to $ex(n,C\_4)$, say $e(G) \geq n^{3/2-\alpha}$; is there a better bound (depending on $\alpha$) in this case? 



Chromatic number of $C_4$-free graphs
For $G$ an $n$ vertex graph which is $C\_4$-free, $\chi(G)=O(\sqrt{n})$, follows from Kővári–Sós–Turán by the argument found [here](https://math.stackexchange.com/questions/2419506/chromatic-number-of-a-graph-with-no-4-cycles) for instance.


Before Johannson proved the chromatic number bound for triangle free graphs, the inequality appeared in a paper of Kim as a conjectured improvement to the girth $>4$ case. In that case, the inequality is due to [Kim](https://pdfs.semanticscholar.org/1af7/3ac8de60158e3fc8f6ed247ea477d4cf1684.pdf) and takes the form


$$\chi(G)≤[1 +o(1)]\frac{\Delta}{\log \Delta},$$


where the $o(1)$ is taken as $\Delta(G)\rightarrow\infty$. For the case that $G$ is $C\_4$ saturated, or nearly so, there is less which appears immediately in a quick search. 


