We show that every n-vertex planar graph is 3-colourable with monochromatic components of size O(n4/9). The best previous bound was O(n1/2) due to Linial, Matou\v{s}ek, Sheffet and Tardos [Combin. Probab. Comput., 2008].
@article{arxiv.2507.03163,
title = {3-Colouring Planar Graphs},
author = {Vida Dujmović and Pat Morin and Sergey Norin and David R. Wood},
journal= {arXiv preprint arXiv:2507.03163},
year = {2025}
}