Exponential patterns in arithmetic Ramsey theory

Abstract

We show that for every finite colouring of the natural numbers there exists a, b > 1 such that the triple {a, b, a b } is monochromatic. We go on to show the partition regularity of a much richer class of patterns involving exponentiation. For example, as a corollary to our main theorem, we

Monochromatic solutions to systems of exponential equations

“…Brown [5], simplifying and extending the proof of Sisto, gave further examples of exponential, monochromatic patterns that are present in an arbitrary 2-colouring and also proved some weaker results for monochromatic patterns in more colours. In [23] we answered Sisto's question by showing that any finite colouring of the positive integers admits a, b > 1 such that a, b, a b are monochromatic and went on to develop, in this context, a theory of patterns defined by compositions of the exponential function. In the present paper we turn from the study of patterns arising as compositions of the exponential function, to understand exponential patterns that arise as solutions to systems of equations.…”