A new ultrafilter proof of Van der Waerden's theorem

TL;DR

This paper introduces a concise ultrafilter-based proof of Van der Waerden's theorem that avoids the use of minimal or idempotent ultrafilters, simplifying the existing proofs.

Contribution

It provides a novel, shorter proof of Van der Waerden's theorem using ultrafilters without relying on minimal or idempotent ultrafilters.

Findings

Establishes the existence of arbitrarily long monochromatic arithmetic progressions.

Simplifies the ultrafilter proof technique for Van der Waerden's theorem.

Avoids complex ultrafilter concepts used in previous proofs.

Abstract

We present a new short proof of Van der Waerden's Theorem about the existence of arbitrarily long monochromatic arithmetic progressions. The proof uses algebra in the compact space of ultrafilters $\beta\N$, but contrarily to the other existing proofs, neither minimal nor idempotent ultrafilters are involved.