On the infinitary van der Waerden Theorem

TL;DR

This paper provides a purely combinatorial proof of the infinitary van der Waerden's theorem, which concerns the existence of monochromatic arithmetic progressions in infinite colorings.

Contribution

It introduces a new combinatorial proof technique for the infinitary van der Waerden's theorem, avoiding analytical or topological methods used previously.

Findings

Established a purely combinatorial proof for the theorem

Confirmed the existence of monochromatic arithmetic progressions in infinite colorings

Simplified understanding of the infinitary van der Waerden's theorem

Abstract

We give a purely combinatorial proof for the infinitary van der Waerden's theorem.