A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for
which the first half of the path is assigned the same sequence of colours as
the second half. The \emph{nonrepetitive chromatic number} of a graph G is
the minimum integer k such that G has a nonrepetitive k-colouring.
Whether planar graphs have bounded nonrepetitive chromatic number is one of the
most important open problems in the field. Despite this, the best known upper
bound is O(n) for n-vertex planar graphs. We prove a O(logn)
upper bound