Exponentiations of ultrafilters

TL;DR

This paper develops foundational theory on how ultrafilters behave under exponentiation, focusing on their combinatorial properties and the existence of idempotent ultrafilters, to address problems in exponential partition regularity.

Contribution

It introduces a general framework for exponentiating ultrafilters, advancing understanding of their combinatorial and algebraic properties in this context.

Findings

Established foundational properties of ultrafilter exponentiation.

Proved existence results for idempotent ultrafilters under exponentiation.

Linked ultrafilter exponentiation to partition regularity of exponential configurations.

Abstract

In recent years, several problems regarding the partition regularity of exponential configurations have been studied in the literature, in some cases using the properties of specific ultrafilters. In this paper, we start to lay down the foundations of a general theory of exponentiations of ultrafilters, with a particular focus on their combinatorial properties and the existence of idempotents.