Using ultrafilters to prove Ramsey-type theorems

TL;DR

This paper introduces ultrafilters as a versatile mathematical tool and demonstrates their application in combinatorics, culminating in a novel ultrafilter-based proof of van der Waerden's theorem.

Contribution

It provides an accessible introduction to ultrafilters and presents a new proof of van der Waerden's theorem using ultrafilters.

Findings

Ultrafilters can be effectively applied in Ramsey theory.

A new ultrafilter-based proof of van der Waerden's theorem is established.

Ultrafilters serve as a powerful tool in combinatorial proofs.

Abstract

Ultrafilters are a tool, originating in mathematical logic and general topology, that has steadily found more and more uses in multiple areas of mathematics, such as combinatorics, dynamics, and algebra, among others. The purpose of this article is to introduce ultrafilters in a friendly manner and present some applications to the branch of combinatorics known as Ramsey theory, culminating with a new ultrafilter-based proof of van der Waerden's theorem.