Conversely, if $h^{(m)}(n)$ is the maximal chromatic number of a graph on $n$ vertices with girth $>m$ then does \[\lim_{n\to \infty}\frac{\log h^{(m)}(n)}{\log n}\] exist, and what is its value?
Conversely, if $h^{(m)}(n)$ is the maximal chromatic number of a graph on $n$ vertices with girth $>m$ then does \[\lim_{n\to \infty}\frac{\log h^{(m)}(n)}{\log n}\] exist, and what is its value?
Erdős [Er59b] proved that \[\lim_{n\to \infty}\frac{\log h^{(m)}(n)}{\log n}\gg \frac{1}{m}\] and, for odd $m$, \[\lim_{n\to \infty}\frac{\log h^{(m)}(n)}{\log n}\leq \frac{2}{m+1},\] and conjectured this is sharp. He had no good guess for the value of the limit for even $m$, other that it should lie in $[\frac{2}{m+2},\frac{2}{m}]$, but could not prove this even for $m=4$.