Translation invariant filters and van der Waerden's Theorem

TL;DR

This paper provides a constructive, axiom-free proof of a strong form of van der Waerden's Theorem using translation invariant filters, demonstrating the existence of many monochromatic arithmetic progressions in finite colorings.

Contribution

It introduces a new, constructive approach employing translation invariant filters to prove a strong version of van der Waerden's Theorem without relying on the axiom of choice.

Findings

Existence of piecewise syndetically-many monochromatic arithmetic progressions

Constructive method for defining maximal filters via recurrence

No use of the axiom of choice or Zorn's Lemma

Abstract

We present a self-contained proof of a strong version of van der Waerden's Theorem. By using translation invariant filters that are maximal with respect to inclusion, a simple inductive argument shows the existence of "piecewise syndetically"-many monochromatic arithmetic progressions of any length k in every finite coloring of the natural numbers. All the presented constructions are constructive in nature, in the sense that the involved maximal filters are defined by recurrence on suitable countable algebras of sets. No use of the axiom of choice or of Zorn's Lemma is needed.