The five-vertex path is nice

Let $d$ be a common exponent large enough to dominate the polynomial width bound in [L1] and the layout theorem [L2]. Fix $ϵ\in (0,\frac{1}{2})$ and a $\overline{P_5}$-free graph $G$ with $\mathrm{\mid }V(G)\mathrm{\mid }\ge {ϵ}^{-10d^2}$, and set $x:={ϵ}^{5d}$. Then $x<2^{-d}$. If $F$ is an induced subgraph of $G$ with $\mathrm{\mid }V(F)\mathrm{\mid }\ge {ϵ}^d\mathrm{\mid }V(G)\mathrm{\mid }$, then $\mathrm{\mid }V(F)\mathrm{\mid }\ge {ϵ}^d{ϵ}^{-10d^2}={ϵ}^{-10d^2+d}\ge {ϵ}^{-5d^2}=x^{-d}\mathrm{.}$ Therefore [L1] applies to $F$ and gives a pure or $x$-sparse $(k,\mathrm{\mid }V(F)\mathrm{\mid }\mathrm{/}{k}^d)$-blockade for some $k\in [2,{ϵ}^{-5d^2}]$.

Applying [L2] to $G$ yields an $({ϵ}^{-1},{ϵ}^{10d^2}\mathrm{\mid }V(G)\mathrm{\mid })$-blockade whose distinct block pairs are either complete or weakly ${ϵ}^d$-sparse. By [L3], this is exactly the niceness condition for $H=P_5$.

Therefore $P_5$ is nice.