Central sets and infinite monochromatic exponential patterns

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.)