3-Colouring Planar Graphs

We show that every $n$-vertex planar graph is 3-colourable with monochromatic components of size $O(n^{4\mathrm{/}9})$. The best previous bound was $O(n^{1\mathrm{/}2})$ due to Linial, Matoušek, Sheffet and Tardos [Combin. Probab. Comput., 2008].