Exponential Patterns in Arithmetic Ramsey Theory

TL;DR

This paper proves that for any finite coloring of natural numbers, certain exponential and multiplicative patterns, including triples and larger sets, always contain monochromatic configurations involving powers and products.

Contribution

It establishes the partition regularity of complex exponential patterns, extending classical Ramsey theory to include exponential structures.

Findings

Existence of monochromatic triples {a,b,a^b} with a,b>1

Partition regularity of exponential pattern classes involving multiple variables

Monochromatic quadruples of the form {a,b,ab,a^b} for any finite coloring

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 $a,b>1$.