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 χ(G)/ω(G), where $χ(G)$ denotes the chromatic number of G and $ω(G)$ the clique number of G. In 1967, Erdős showed that \[ ( 1/4 +o(1) ) n/(log_2 n)^2 ≤ f(n) ≤ ( 4+o(1) ) n/(log_2 n)^2 .\] We show that \[ f(n) ≤ (c+o(1)) n/(log_2 n)^2\] 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.