Polynomial Gyárfás–Sumner conjecture

Let $G$ be a finite graph, let $T$ be a forest, and write $\chi(G)$ for the chromatic number and $\omega(G)$ for the clique number. Polynomial Gyárfás–Sumner conjecture. For every forest $T$, there exists an integer $d\ge2$ such that

$$\chi(G)\le\omega(G)^d$$

for all $T$-free graphs $G$. This is a polynomial strengthening of the Gyárfás–Sumner conjecture. Most known cases have weaker, often super-exponential, bounds; the conjecture remains open, with $P_5$ identified in the source as the smallest open case.