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

Erdos 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 e greater than or equal to 3 and k epsilon N there exist constants C-1 and C-2 so that almost all graphs on n vertices and m edges whose girth is greater than e have chromatic number at least k, provided that C(1)n less than or equal to m less than or equal to C(2)n(e/(e-1)). (C) 2001 John Wiley & Sons, Inc.