Rado's criterion over squares and higher powers

2018/06/13 by Sam Chow, Chow, Sam, Sofia Lindqvist +3 · 1 citation
Mathematics · #11B30 #11D72 #11L15 #Combinatorics (math.CO) #FOS: Mathematics #Functional Equations Stability Results #History and Theory of Mathematics #Mathematical and Theoretical Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1806.05002

openalex publication_date 2018/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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))klog 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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