# Rado's criterion over squares and higher powers

### Sam Chow

University of Oxford, UK### Sofia Lindqvist

University of Oxford, UK### Sean Prendiville

University of Manchester, UK

## 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.

## Cite this article

Sam Chow, Sofia Lindqvist, Sean Prendiville, Rado's criterion over squares and higher powers. J. Eur. Math. Soc. 23 (2021), no. 6, pp. 1925–1997

DOI 10.4171/JEMS/1047