A hole in a graph is an induced subgraph which is a cycle of length at least four. We prove that for every positive integer k, every triangle-free graph with sufficiently large chromatic number contains holes of k consecutive lengths.
@article{arxiv.1509.06563,
title = {Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes},
author = {Alex Scott and Paul Seymour},
journal= {arXiv preprint arXiv:1509.06563},
year = {2018}
}