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

TL;DR

This work delivers an elementary, van der Waerden-free proof of the sufficiency part of Rado's theorem for partition regularity of linear Diophantine equations. It develops and leverages a toolkit of shifts, thick/syndetic and piecewise syndetic sets, Delta-sets, Banach density, and the compactness principle to circumvent advanced machinery. The core strategy reduces a general linear form $P(x_1,\ldots,x_k)=\sum c_i x_i$ with Rado's condition to a three-variable auxiliary form $Q(y_1,y_2,y_3)=c y_1-c y_2-d y_3$ and uses joint partition regularity in the setting of multiplicatively piecewise syndetic sets to obtain monochromatic solutions, subsequently extending this to $P$ via $P_q$-type transformations and a compactness argument. The result fills a longstanding gap by providing a self-contained, elementary proof and suggests avenues for extending the approach to systems and injective solutions.

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.

Theorems & Definitions (42)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: Schur
• Theorem 1.4: van der Waerden
• Theorem 1.5: van der Waerden -- Brauer
• Definition 1.6
• Theorem 1.7: Rado
• Definition 2.1
• Proposition 2.2
• Proposition 2.3