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 th powers provided the equation has at least log 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.
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Sam Chow, Sofia Lindqvist, Sean Prendiville, Rado's criterion over squares and higher powers. J. Eur. Math. Soc. 23 (2021), no. 6, pp. 1925–1997DOI 10.4171/JEMS/1047