Monochromatic Solutions to Systems of Exponential Equations

Abstract

Let $n\in \mathbb{N}$, $R$ be a binary relation on $[n]$, and $C_1(i,j),\dots ,C_n(i,j)\in \mathbb{Z}$, 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 $X_1,\dots ,X_n,Y_1,\dots ,Y_n$. The aim of this paper is to classify precisely which of these systems admit a monochromatic solution ($X_i,Y_i\ne 1)$ in an arbitrary finite colouring of the natural numbers. This result could be viewed as an analogue of Rado's theorem for exponential patterns.