chromatic number of a graph versus its complement

What can be said about the rate of growth of f(n), defined by

$f(n)=\underset{|V(G)|=n}{min}[\chi (G)+\chi (\overline{G})],$

where the minimum is taken over all graphs G on n vertices.

Two observations.

(1) Either G or $\overline{G}$ contains a clique on roughly logn vertices by Ramsey theory, so $f(n)\ge {c}_{1}\mathrm{log}n$ for some constant ${c}_{1}>0$.

(2) If G=G(n,1/2) is a random graph, then $\chi (G)\approx \chi (\overline{G})\approx n/\mathrm{log}n$ n/ log n almost surely, so we also have $f(n)\le {c}_{2}\phantom{\rule{thinmathspace}{0ex}}n/\mathrm{log}n$ n/log n for some constant ${c}_{2}>0$.

These bounds seem hopelessly far apart.

Can we improve on the bounds

${c}_{1}\mathrm{log}n\le f(n)\le {c}_{2}\phantom{\rule{thinmathspace}{0ex}}n/\mathrm{log}n$

for all sufficiently large n?

What can be said about the rate of growth of f(n), defined by

$f(n)=\underset{|V(G)|=n}{min}[\chi (G)+\chi (\overline{G})],$

where the minimum is taken over all graphs G on n vertices.

Two observations.

(1) Either G or $\overline{G}$ contains a clique on roughly logn vertices by Ramsey theory, so $f(n)\ge {c}_{1}\mathrm{log}n$ for some constant ${c}_{1}>0$.

(2) If G=G(n,1/2) is a random graph, then $\chi (G)\approx \chi (\overline{G})\approx n/\mathrm{log}n$ n/ log n almost surely, so we also have $f(n)\le {c}_{2}\phantom{\rule{thinmathspace}{0ex}}n/\mathrm{log}n$ n/log n for some constant ${c}_{2}>0$.

These bounds seem hopelessly far apart.

Can we improve on the bounds

${c}_{1}\mathrm{log}n\le f(n)\le {c}_{2}\phantom{\rule{thinmathspace}{0ex}}n/\mathrm{log}n$

for all sufficiently large n?