Path Ramsey number for random graphs

Answering a question raised by Dudek and Pra\l{}at, we show that if $pn\to \mathrm{\infty }$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there exists a monochromatic path of length $n(2\mathrm{/}3+o(1))$. This result is optimal in the sense that $2\mathrm{/}3$ 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 $(1-ϵ)\left(\genfrac{}{}{0px}{}{n}{2}\right)$ edges, whenever $G$ is $2$-edge-coloured, there is a monochromatic path of length at least $(2\mathrm{/}3-100\sqrt{ϵ})n$. 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 $2n\mathrm{/}3$.