Gyárfás's conjecture on chi-boundedness of forest-free graphs

Let $H$ be a fixed forest, and let $\mathcal{G}(H)$ denote the hereditary class of graphs with no induced subgraph isomorphic to $H$. A graph class is chi-bound if there is a function $f$ such that $\chi(G)\leq f(\omega(G))$ for every graph $G$ in the class, where $\chi(G)$ is the chromatic number and $\omega(G)$ is the clique number. Gyárfás's conjecture. The class $\mathcal{G}(H)$ is chi-bound for every fixed forest $H$. This conjecture asks whether forbidding any fixed induced forest yields a bounded relationship between chromatic number and clique number; it is presented as an open problem in the source's discussion of Gyárfás's conjectures.