Scott, AlexSeymour, PaulSpirkl, Sophie2022-08-122022-08-122022-10https://doi.org/10.1002/jgt.22862http://hdl.handle.net/10012/18534This is the peer reviewed version of the following article: Scott, A., Seymour, P., & Spirkl, S. (2022). Polynomial bounds for chromatic number. III. Excluding a double star. Journal of Graph Theory, 101(2), 323–340. https://doi.org/10.1002/jgt.22862, which has been published in final form at https://doi.org/10.1002/jgt.22862. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.A “double star” is a tree with two internal vertices. It is known that the Gyárfás-Sumner conjecture holds for double stars, that is, for every double star H, there is a function fH such that if G does not contain H as an induced subgraph then x(G) ≤ fH(w(G)) (where x, w are the chromatic number and the clique number of G). Here we prove that fH can be chosen to be a polynomial.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalpolynomial boundschromatic numberdouble starArticle