  Bal and DeBiasio [Partitioning random graphs into monochromatic components,
Electron. J. Combin. 24 (2017), Paper 1.18] put forward a conjecture concerning
the threshold for the following Ramsey-type property for graphs $G$: every
$k$-colouring of the edge set of $G$ yields $k$ pairwise vertex disjoint
monochromatic trees that partition the whole vertex set of $G$. We determine
the threshold for this property for two colours.
