Bull-free graphs and $\chi$-boundedness

A bull is a graph obtained from a four-vertex path by adding a vertex adjacent to the two middle vertices of the path. A graph $G$ is bull-free if no induced subgraph of $G$ is a bull. We prove that for all $k,t\in \mathbb{N}$, if $G$ is a bull-free graph of clique number at most $k$ and every triangle-free induced subgraph of $G$ has chromatic number at most $t$, then $G$ has chromatic number at most $k^{\mathcal{O}(\log t)}$. We further show that the bound $k^{\mathcal{O}(\log t)}$ is best possible up to a multiplicative constant in the exponent. The previously best known bound was of the form $2^{poly(k,t)\log poly(k,t)}$, due to Thomass\'{e}, Trotignon and Vu\v{s}kovi\'{c} (2017), whose proof uses Chudnovsky's decomposition theorem for bull-free graphs. Our result also implies that every $\chi$-bounded hereditary class of bull-free graphs is polynomially $\chi$-bounded. This was recently given a 10-page proof by Chudnovsky, Cook, Davies and Oum (2023) that relies heavily on Chudnovsky's structure theorem. Our proof is a single page long and completely avoids the structure theorem, using only one result of Chudnovsky and Safra (which itself has a short proof).