Coloured solutions of equations in finite groups

In this article, we consider the relations between colourings and some equations in finite groups. We will express relations linking the numbers of the differently coloured solutions of an equation that depend only on the cardinality of the colouring and not on the distribution of the colours. This gives a link between Ramsey theory that investigates the existence of monochromatic solutions and what is now called anti-Ramsey theory that investigates the existence of rainbow solutions. Both theories are in expansion. We will apply these results to the counting of rainbow 3-term arithmetic progressions in any abelian group with equinumerous three-colouring and to the counting of points on a conic defined on a finite field. We will end by discussing the generalised case of a system of equations.