  We investigate the linear chromatic number $\chi_{\text{lin}}(G(n,p))$ of the
binomial random graph $G(n,p)$ on $n$ vertices in which each edge appears
independently with probability $p=p(n)$. For dense random graphs ($np \to
\infty$ as $n \to \infty$), we show that asymptotically almost surely
$\chi_{\text{lin}}(G(n,p)) \ge n (1 - O( (np)^{-1/2} ) ) = n(1-o(1))$.
Understanding the order of the linear chromatic number for subcritical random
graphs ($np < 1$) and critical ones ($np=1$) is relatively easy. However,
supercritical sparse random graphs ($np = c$ for some constant $c > 1$) remain
to be investigated.
