Triangle-free graphs with large chromatic numbers

It is shown that there are two positive constants c1,c2 such that the maximum possible chromatic number of a triangle-free graph with m>1 edges is at most c1m1/3/(log m)2/3 and at least c2m1/3/(log m)2/3. This is deduced from results of Ajtai, Komlós, Szemerédi, Kim and Johansson, and settles a problem of Erdős.