Improved upper bounds for the Poljak–Rödl function

Improve the upper bound f(m) ≤ ⌈m/2⌉ + 3 for the Poljak–Rödl function and determine whether lim sup f(n)/n = 0.

Background

The best stated general upper bound is f(m) ≤ ⌈m/2⌉ + 3. The authors explain that improving this bound may require proving stronger chromatic lower bounds for exponential graphs of the form K_n^{Ωw(K{m'})}. They also ask whether the ratio f(n)/n tends to zero along the upper limit.