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arXiv · 1806.05002

Rado's criterion over squares and higher powers

Sam Chow·Sofia Lindqvist·Sean Prendiville
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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 $k$th powers, provided the equation has at least $(1+o(1))k\log k$ 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. 2018-06-13. Rado's criterion over squares and higher powers. https://arxiv.org/abs/1806.05002

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