Let $L_1, \dots , L_n$ be finite sets of lines in $\mathbb{F}^n$ where $\mathbb{F}$ is any field. A {\em multijoint} is a point of $\mathbb{F}^n$ which is at the intersection of a line from each family in such a way that the directions of the lines span. Call the set of multijoints $J$.
We prove (subject to a technical hypothesis) that it is possible to $n$-colour the multijoints in such a way that every line contains at most $O_n(|J|^{1/n})$ of its own colour. We explain with reference to problems from harmonic analysis why this is the natural result in this context.
This is joint work with Stefan Valdimarsson.
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