Fractionally colouring $P_5$-free graphs

We obtain some $d\ge2$ such that every graph $G$ with no induced copy of the five-vertex path $P_5$ has at most $\alpha(G)\omega(G)^d$ vertices. This ``off-diagonal Ramsey'' statement implies that every such graph $G$ has fractional chromatic number at most $\omega(G)^d$, and is another step towards the polynomial Gy\'arf\'as-Sumner conjecture for $P_5$. The proof uses the recent Erd\H{o}s-Hajnal result for $P_5$ and adapts a decomposition argument for $P_5$-free graphs developed by the author in an earlier paper.