Graphs with large chromatic number induce $3k$-cycles
  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. 
Marthe Bonamy, Pierre Charbit, St\'ephan Thomass\'e
cs.DM math.CO