Partitioning by Monochromatic Trees

Anyr-edge-colouredn-vertex complete graphKncontains at mostrmonochromatic trees, all of different colours, whose vertex sets partition the vertex set ofKn, providedn⩾3r4r! (1−1/r)3(1−r) log r. This comes close to proving, for largen, a conjecture of Erdős, Gyárfás, and Pyber, which states thatr−1 trees suffice for alln.