3-Colouring Graphs Excluding a Fixed Minor

We show that, for every fixed graph $H$, every $n$-vertex graph $G$ that excludes $H$ as a minor is $3$-colourable with clustering $O_H(n^{4\mathrm{/}9})$. That is, there exists a function $f$ such that for every graph $H$, every $n\ge 1$, every $n$-vertex graph $G$ that excludes $H$ as a minor has a vertex colouring with $3$ colours in which each monochromatic component has size at most $f(H)\cdot n^{4\mathrm{/}9}$. This generalizes a recent result of Dujmović, Morin, Norin, and Wood ($\mathit{\text{arXiv}}$:2507.03163) from planar graphs to all proper minor-closed graph classes and is the first improvement on clustered $3$-colouring of proper minor-closed graph classes since the upper bound of $O_H(\sqrt{n})$ due to Linial, Matoušek, Sheffet, and Tardos ($\mathit{\text{Comb.}}\mathit{\text{Prob.}}\mathit{\text{Comput.}}$, $\mathbf{\text{17}}$(4):577–589, 2008).