Central sets and infinite monochromatic exponential patterns

TL;DR

This paper demonstrates that in any finite coloring of natural numbers, there exists an infinite sequence where all exponential configurations, including towers of exponents, are monochromatic, using properties of central sets.

Contribution

It introduces a novel application of central sets to establish the existence of infinite monochromatic exponential patterns in finite colorings.

Findings

Existence of infinite sequences with monochromatic exponential towers.

Extension of Hindman's theorem to exponential configurations.

Utilization of central set properties in combinatorial number theory.

Abstract

We use the combinatorial properties of central sets to prove a result about the existence of exponential monochromatic patterns, in the style of Hindman's Finite Sums Theorem. More precisely, we prove that for every finite coloring of the natural numbers there exists an infinite sequence such that all suitable exponential configurations originating from its distinct elements are monochromatic, including towers of exponentiations. (Some restrictions apply on the order in which elements are considered.)