LAUSR

When H is a forest, the Gyárfás-Sumner conjecture implies that every graph G with no induced subgraph isomorphic to H and with bounded clique number has a stable set of linear size. We cannot prove that, but we prove that every such graph G has a stable set of size |G|1-o(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|G|^{1-o(1)}$$\end{document}. If H is not a forest, there need not be such a stable set. Second, we prove that when H is a “multibroom”, there is a stable set of linear size. As a consequence, we deduce that all multibrooms satisfy a “fractional colouring” version of the Gyárfás-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting.