Abstract
Any r -edge-coloured n -vertex complete graph K n contains at most r monochromatic trees, all of different colours, whose vertex sets partition the vertex set of K n , provided n ⩾3 r 4 r ! (1−1/ r ) 3(1− r ) log r . This comes close to proving, for large n , a conjecture of Erdős, Gyárfás, and Pyber, which states that r −1 trees suffice for all n .
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