Graphs with large chromatic number induce 3k-cycles
Résumé
Answering a question of Kalai and Meshulam, we prove that graphs without induced cycles of length 3k have bounded chromatic number. This implies the very first case of a much broader question asserting that every graph with large chromatic number induces a graph H such that the sum of the Betti numbers of the independence complex of H is also large.