Almost all graphs with high girth and suitable density have high chromatic number

Erdős proved that there exist graphs of arbitrarily high girth and arbitrarily high chromatic number. We give a different proof (but also using the probabilistic method) that also yields the following result on the typical asymptotic structure of graphs of high girth: for all ` ≥ 3 and k ∈ N there exist constants C1 and C2 so that almost all graphs on n vertices and m edges whose girth is greater than ` have chromatic number at least k, provided that C1n ≤ m ≤ C2n.