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 show that for every $n \in \mathbb{N}$ and for every finite colouring of the natural numbers, we may find a monochromatic set including the integers $x_1,\ldots,x_n >1$; all products of distinct $x_i$; and all "exponential compositions" of distinct $x_i$ which respect the order $x_1,\ldots,x_n$. In particular, for every finite colouring of the natural numbers one can find a monochromatic quadruple of the form $\{ a,b,ab,a^b \}$, where $a,b>1$.