We investigate the problem of determining how many monochromatic trees are necessary to cover the vertices of an edge-coloured random graph. More precisely, we show that for $pgg n^{-1/6}{(ln n)}^{1/6}$, in any $3$-edge-colouring of the random graph $G(n,p)$ we can find three monochromatic trees such that their union covers all vertices. This improves, for three colours, a result of Bucic, Korandi and Sudakov.
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