Fractionally colouring $P_5$-free graphs
Tung H. Nguyen
TL;DR
The paper establishes an off-diagonal Ramsey-type bound for $P_5$-free graphs by showing there exists $d\ge9$ (indeed $d\ge2$ is the target in the main theorem, with subsequent refinements) such that any $P_5$-free graph $G$ satisfies $|G| \le \alpha(G)\omega(G)^d$, equivalently $\chi^*(G) \le \omega(G)^d$. The approach fuses the Erdős–Hajnal property for $P_5$ with a density-increment/decomposition framework based on high-$\rho$ (Hall ratio) structures, including complete pairs, anticomplete pairs, and blockades, and leverages Rödl-type results to handle sparse/dense cases. A central component is a robust iterative procedure that extracts anticomplete pairs with increasingly large $\rho$-values, leading to a global bound and, importantly, to closure properties under disjoint union for poly-$\chi^*$- and poly-$\rho$-bounding as well as off-diagonal Erdős–Hajnal. These results connect off-diagonal Ramsey-type phenomena to polynomial bounds on fractional chromatic numbers for $P_5$-free graphs, contributing to the broader program toward a polynomial Gyárfás–Sumner conjecture for forests.
Abstract
We obtain some $d\ge2$ such that every graph $G$ with no induced copy of the five-vertex path $P_5$ has at most $α(G)ω(G)^d$ vertices. This ``off-diagonal Ramsey'' statement implies that every such graph $G$ has fractional chromatic number at most $ω(G)^d$, and is another step towards the polynomial Gyárfás-Sumner conjecture for $P_5$. The proof uses the recent Erdős-Hajnal result for $P_5$ and adapts a decomposition argument for $P_5$-free graphs developed by the author in an earlier paper.