A Van der Waerden-free proof of Rado's theorem

TL;DR

This paper provides an elementary proof of Rado's theorem on partition regularity of linear equations, avoiding the use of van der Waerden's theorem and relying on basic combinatorial principles.

Contribution

It introduces a new proof that is elementary and does not depend on van der Waerden's theorem, simplifying the understanding of Rado's theorem.

Findings

Proof of Rado's theorem without van der Waerden's theorem

Uses only fundamental combinatorial properties and compactness

Simplifies the conceptual framework of partition regularity

Abstract

We present a proof of the sufficiency of Rado's condition for the partition regularity of linear Diophantine equations that avoids any use of van der Waerden's theorem. The proof is based on fundamental properties that are common knowledge in combinatorics of numbers and is entirely elementary, with the sole exception of a standard application of the compactness principle.