Carbonero, AlvaroHompe, PatrickMoore, BenjaminSpirkl, Sophie2022-09-202022-09-202023-01https://doi.org/10.1016/j.jctb.2022.09.001http://hdl.handle.net/10012/18757The final publication is available at Elsevier via https://doi.org/10.1016/j.jctb.2022.09.001 © 2023. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We prove that for every n, there is a graph G with χ(G) ≥ n and ω(G) ≤ 3 such that every induced subgraph H of G with ω(H) ≤ 2 satisfies χ(H) ≤ 4.This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/χ-Boundednessinduced subgraphdirected graphArticle