A note on the maximum ratio between chromatic number and clique number

Let $f(n)$ be the maximum, over all graphs $G$ on $n$ vertices, of the ratio $\frac{\chi (G)}{\omega (G)}$, where $\chi (G)$ denotes the chromatic number of $G$ and $\omega (G)$ the clique number of $G$. In 1967, Erdős showed that $$(\frac{1}{4}+o(1))\frac{n}{({\log }_{2}n{)}^2}\le f(n)\le (4+o(1))\frac{n}{({\log }_{2}n{)}^2}\mathrm{.}$$ We show that $$f(n)\le (c+o(1))\frac{n}{({\log }_{2}n{)}^2}$$ for some $c<3.72$. This follows from recent improvements in the asymptotics of Ramsey numbers and is the first improvement in the asymptotics of $f(n)$ established by Erdős.