  Akbari, Etesami, Mahini, and Mahmoody conjectured that every proper edge
colouring of $K_n$ with $n$ colours contains a Hamilton cycle with $\leq O(\log
n)$ colours. They proved that there is always a Hamilton cycle with $\leq
8\sqrt n$ colours. In this note we improve this bound to $O(\log^3 n)$.
