Generalised Rado and Roth criteria

TL;DR

This paper characterizes when polynomial equations of a certain form have monochromatic solutions under any finite coloring, and establishes conditions for solutions in dense sets, extending classical results in Ramsey theory and additive combinatorics.

Contribution

It provides a complete characterization of equations admitting monochromatic solutions based on Rado's criterion and intersectivity, extending classical results to polynomial equations of higher degree.

Findings

Rado's criterion and intersectivity fully characterize monochromatic solutions.

A Roth-type theorem for polynomial equations with conditions on the sum of coefficients.

Sharp asymptotic bounds for the number of solutions in monochromatic or dense sets.

Abstract

We study the Ramsey properties of equations $a_1P(x_1) + \cdots + a_sP(x_s) = b$, where $a_1,\ldots,a_s,b$ are integers, and $P$ is an integer polynomial of degree $d$. Provided there are at least $(1+o(1))d^2$ variables, we show that Rado's criterion and an intersectivity condition completely characterise which equations of this form admit monochromatic solutions with respect to an arbitrary finite colouring of the positive integers. Furthermore, we obtain a Roth-type theorem for these equations, showing that they admit non-constant solutions over any set of integers with positive upper density if and only if $b= a_1 + \cdots + a_s = 0$. In addition, we establish sharp asymptotic lower bounds for the number of monochromatic/dense solutions (supersaturation).