Rainbow solutions of linear equations over ZpZp

We prove that if the group ZpZp, with p   a prime, is coloured with k⩾4k⩾4 different colours such that each colour appears at least k   times, then for any a1,…,ak,ba1,…,ak,b in ZpZp with not all the aiai being equal, we may solve the equation a1x1+⋯+akxk=ba1x1+⋯+akxk=b so that each of the variables is chosen in a different colour class. This generalises a similar result concerning three colour classes due to Jungić, Licht, Mahdian, Nešetřil and Radoičić.In the course of our proof we classify, with some size caveats, the sets in ZpZp which satisfy the inequality |A1+⋯+An|⩽|A1|+⋯+|An||A1+⋯+An|⩽|A1|+⋯+|An|. This is a generalisation of an inverse theorem due to Hamidoune and Rødseth concerning the case n=2n=2.