  An edge-coloring of a connected graph $G$ is called a {\it monochromatic
connection coloring} (MC-coloring, for short), introduced by Caro and Yuster,
if there is a monochromatic path joining any two vertices of the graph $G$. Let
$mc(G)$ denote the maximum number of colors used in an MC-coloring of a graph
$G$. Note that an MC-coloring does not exist if $G$ is not connected, and in
this case we simply let $mc(G)=0$. We use $G(n,p)$ to denote the
Erd\"{o}s-R\'{e}nyi random graph model, in which each of the $\binom{n}{2}$
pairs of vertices appears as an edge with probability $p$ independently from
other pairs. For any function $f(n)$ satisfying $1\leq f(n)<\frac{1}{2}n(n-1)$,
we show that if $\ell n \log n\leq f(n)<\frac{1}{2}n(n-1)$ where $\ell\in
\mathbb{R}^+$, then $p=\frac{f(n)+n\log\log n}{n^2}$ is a sharp threshold
function for the property $mc\left(G\left(n,p\right)\right)\ge f(n)$; if
$f(n)=o(n\log n)$, then $p=\frac{\log n}{n}$ is a sharp threshold function for
the property $mc\left(G\left(n,p\right)\right)\ge f(n)$.
