Ultrafilters, monads and combinatorial properties

TL;DR

This paper applies nonstandard hyperextension methods to ultrafilter monads, advancing Ramsey theory by analyzing tensor products and combinatorial properties, extending previous techniques to broader set contexts.

Contribution

It introduces a novel nonstandard approach to ultrafilter monads, enabling detailed analysis of tensor products and combinatorial properties in Ramsey theory.

Findings

Characterization of ultrafilter monads via nonstandard methods

Extension of partition regular Diophantine equation techniques

Multiple applications demonstrated through examples

Abstract

We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultrafilters. This is performed by studying in detail arbitrary tensor products of ultrafilters, as well as by characterizing their combinatorial properties by means of their monads. This extends to arbitrary sets methods previously used to study partition regular Diophantine equations on \N. Several applications are described by means of multiple examples.