We resolve the Ramsey problem for @math for all polynomials @math over @math .
Suppose that @math is coloured with @math colours. Then there is some colour class containing at least @math quadruples of the form @math . We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado's characterisation of a partition regular equation remains valid over the set of positive @math th powers, provided the equation has at least @math variables. We thus completely describe which diagonal forms are partition regular and which are not, given sufficiently many variables. In addition, we prove a supersaturated version of Rado's theorem for a linear equation restricted either to squares minus one or to logarithmically-smooth numbers. 