Cycles in triangle-free graphs of large chromatic number

More than twenty years ago Erdős conjectured [4] that a triangle-free graph G of chromatic number k ≥ k0(ε) contains cycles of at least k2−ε different lengths as k →∞. In this paper, we prove the stronger fact that every triangle-free graph G of chromatic number k ≥ k0(ε) contains cycles of 1 64 (1− ε)k 2 log k4 consecutive lengths, and a cycle of length at least 14 (1− ε)k 2 log k. As there exist triangle-free graphs of chromatic number k with at most roughly 4k log k vertices for large k, these results are tight up to a constant factor. We also give new lower bounds on the circumference and the number of different cycle lengths for k-chromatic graphs in other monotone classes, in particular, for Kr-free graphs and graphs without odd cycles C2s+1.