Gallai 3-colourings of random graphs

A Gallai $k$-colouring of a graph $G$ is a colouring of $E(G)$ with $k$ colours that induces no rainbow triangles, that is, a triangle with edges of 3 different colours. We give a first step towards estimating the number of Gallai colourings of the Erd\H{o}s-R\'enyi random graph, by proving that for every $\delta > 0$ there are $c$ and $C$ such that with high probability the number of Gallai 3-colourings of $G(n,p)$ is at least $3^{(1-\delta)\binom{n}{2}p}$ for $p \leq cn^{-1/2}$, and at most $2^{(1+\delta)\binom{n}{2}p}$ for $p \geq Cn^{-1/2}$.