  Answering a question raised by Dudek and Pra\l{}at, we show that if
$pn\rightarrow \infty$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there
exists a monochromatic path of length $n(2/3+o(1))$. This result is optimal in
the sense that $2/3$ cannot be replaced by a larger constant.
  As part of the proof we obtain the following result which may be of
independent interest. We show that given a graph $G$ on $n$ vertices with at
least $(1-\epsilon)\binom{n}{2}$ edges, whenever $G$ is $2$-edge-coloured,
there is a monochromatic path of length at least $(2/3-100\sqrt{\epsilon})n$.
This is an extension of the classical result by Gerencs\'er and Gy\'arf\'as
which says that whenever $K_n$ is $2$-coloured there is a monochromatic path of
length at least $2n/3$.
