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A new ultrafilter proof of Van der Waerden's theorem

Mauro Di Nasso

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 $β\N$, but contrarily to the other existing proofs, neither minimal nor idempotent ultrafilters are involved.

A new ultrafilter proof of Van der Waerden's theorem

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 , but contrarily to the other existing proofs, neither minimal nor idempotent ultrafilters are involved.
Paper Structure (2 sections, 4 theorems, 6 equations)

This paper contains 2 sections, 4 theorems, 6 equations.

Table of Contents

  1. Preliminaries
  2. The ultrafilter proof

Key Result

Theorem 1.1

Let $\mathcal{G}\subseteq\mathcal{P}$ be a family of patterns. The following two properties are equivalent:

Theorems & Definitions (8)

  • Theorem 1.1
  • proof
  • Lemma 1.2
  • Lemma 1.3
  • proof
  • Theorem 2.1: Van der Waerden
  • proof
  • Remark 2.2