Tony Carbery University of Edinburgh Let L1,…,LnL1,…,Ln be finite sets of lines in FnFn where FF is any field. A {\em multijoint} is a point of FnFn 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 JJ. We prove (subject to a technical hypothesis) that it is possible to nn-colour the multijoints in such a way that every line contains at most On(|J|1/n)On(|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.