  For a graph $G$, let $\chi(G)$ and $\omega(G)$ respectively denote the
chromatic number and clique number of $G$. We give an explicit structural
description of ($P_5$,gem)-free graphs, and show that every such graph $G$
satisfies $\chi(G)\le \lceil\frac{5\omega(G)}{4}\rceil$. Moreover, this bound
is best possible.
