Monochromatic Solutions to Systems of Exponential Equations

TL;DR

This paper classifies which systems of exponential equations have monochromatic solutions in any finite coloring of natural numbers, extending Rado's theorem to exponential patterns.

Contribution

It provides a precise classification of exponential systems that admit monochromatic solutions, analogous to Rado's theorem for linear equations.

Findings

Characterization of exponential systems with monochromatic solutions

Extension of Rado's theorem to exponential equations

Conditions for the existence of solutions in finite colorings

Abstract

Let $n\in \mathbb{N}$, $R$ be a binary relation on $[n]$, and $C_1(i,j),\ldots,C_n(i,j) \in \mathbb{Z}$, for $i,j \in [n]$. We define the exponential system of equations $\mathcal{E}(R,(C_k(i,j)_{i,j,k})$ to be the system \[ X_i^{Y_1^{C_1(i,j)} \cdots Y_n^{C_n(i,j)} } = X_j , \text{ for } (i,j) \in R ,\] in variables $X_1,\ldots,X_n,Y_1,\ldots,Y_n$. The aim of this paper is to classify precisely which of these systems admit a monochromatic solution ($X_i,Y_i \not=1)$ in an arbitrary finite colouring of the natural numbers. This result could be viewed as an analogue of Rado's theorem for exponential patterns.