Monochromatic solutions to systems of exponential equations

Let n∈N, R be a binary relation on [n], and C1(i,j),…,Cn(i,j)∈Z, for i,j∈[n]. We define the exponential system of equations E(R,(Ck(i,j)i,j,k) to be the systemXiY1C1(i,j)⋯YnCn(i,j)=Xj, for (i,j)∈R, in variables X1,…,Xn,Y1,…,Yn. The aim of this paper is to classify precisely which of these systems admit a monochromatic solution (Xi,Yi≠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.