MathNet 2018-part-02 — 181 problems (EN–VI) Question 159

Patricia placed 2018 points in the plane so that the distances between any 2 points are pairwise distinct. She then colours each of her 2018 points, taking care that, for each point $P$, the points $Q$ and $R$ placed closest to and farthest from $P$ have the same colour as $P$.
At most how many colours can Patricia have used?