Gyárfás–Sumner conjecture on chi-bounded forest-free graphs

Let $T$ be a forest, and let a graph be $T$-free if it has no induced subgraph isomorphic to $T$. A hereditary graph class is chi-bounded if there is a function $f:\mathbb{N}\to\mathbb{N}$ such that every graph $G$ in the class and every induced subgraph $H$ of $G$ satisfy $\chi(H)\leqslant f(\omega(H))$, where $\chi(H)$ is the chromatic number and $\omega(H)$ is the clique number. Gyárfás–Sumner conjecture. For every forest $T$, the class of $T$-free graphs is chi-bounded. The conjecture was disproved by Briański, Davies, and Walczak, who constructed chi-bounded classes that are not polynomially chi-bounded.