We resolve the Ramsey problem for @math for all polynomials @math over @math .
Ramsey theory has witnessed exciting development recently. We refer the readers to the papers of Green and Sanders @cite_0 and of Moreira @cite_2 for the problem involving sum and product of @math and @math , and to the paper of Chow, Lindqvist and Prendiville @cite_13 for generalisation of Rado’s criterion to higher powers for sufficiently many variables.
@cite_0 @cite_13 @cite_2
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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.