Abstract :
We prove that if the group image, with p a prime, is coloured with image different colours such that each colour appears at least k times, then for any image in image with not all the image being equal, we may solve the equation image 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 image which satisfy the inequality image. This is a generalisation of an inverse theorem due to Hamidoune and Rødseth concerning the case image.
