Coloring graphs with no long induced path

Source abstract

Let $P_t$ denote the induced path on $t$ vertices. Let $ω(G)$ denote the maximum number of vertices in a clique of a graph $G$. Previously Gyárfás (1987) proved that every $P_t$-free graph $G$ satisfies $χ(G)\le(t-1)^{ω(G)-1}$, and Gravier, Hoàng, and Maffray (2003) improved this to $χ(G)\le (t-2)^{ω(G)-1}$ for $t\ge4$. We prove that for $t\ge5$,every $P_t$-free graph $G$ satisfies $χ(G)<c_tλ_t^{ω(G)-1}$, where $λ_t=\tfrac12\bigl(t-2+\sqrt{t(t-4)}\bigr)<t-2$ and $c_t=\sqrt{1+4/(t(t-4))}=1+O(t^{-2})$ as $t\to\infty$. The proof is based on a refinement of the Gyárfás path argument and was found by Claude Fable 5.1 of Anthropic.