  Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it
contains no induced subgraph isomorphic to $H_1$ or $H_2$. A $P_t$ is the path
on $t$ vertices. A chair is a $P_4$ with an additional vertex adjacent to one
of the middle vertices of the $P_4$. A graph $G$ is $k$-vertex-critical if $G$
has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic
number less than $k$. In this paper, we prove that there are finitely many
5-vertex-critical $(P_5,chair)$-free graphs.
