  Finding families that admit a linear $\chi$-binding function is a problem
that has interested researchers for a long time. Recently, the question of
finding linear subfamilies of $2K_2$-free graphs has garnered much attention.
In this paper, we are interested in finding a linear subfamily of a specific
superclass of $2K_2$-free graphs, namely $(P_3\cup P_2)$-free graphs. We show
that the class of $\{P_3\cup P_2,gem\}$-free graphs admits $f=2\omega$ as a
linear $\chi$-binding function. Furthermore, we give examples to show that the
optimal $\chi$-binding function $f^*\geq
\left\lceil\frac{5\omega(G)}{4}\right\rceil$ for the class of $\{P_3\cup P_2,
gem\}$-free graphs and that the $\chi$-binding function $f=2\omega$ is tight
when $\omega=2$ and $3$.
