Rado's criterion over squares and higher powers

Chow, Sam and Lindqvist, Sofia and Prendiville, Sean (2021) Rado's criterion over squares and higher powers. Journal of the European Mathematical Society, 23 (6). pp. 1925-1997. ISSN 1435-9855

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Abstract

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 kth powers, provided the equation has at least (1+o(1))klogk 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.