Van der Waerden's theorem on arithmetic progressions -- a survey of some historical and modern developments

TL;DR

This survey reviews the historical origins, connections, and significant developments inspired by Van der Waerden's theorem in Ramsey theory since its publication in 1927.

Contribution

It provides a comprehensive overview of the theorem's historical context, its influence across research areas, and key modern results it has inspired.

Findings

Highlights the theorem's role as a catalyst in Ramsey theory

Traces its historical development and influence

Summarizes important modern results inspired by the theorem

Abstract

Van der Waerden's theorem, published in Nieuw. Arch. Wisk. 15 (1927), acted as a catalyst for major further developments in Ramsey theory. In this survey, we delve into the legacy of this mathematical gem, tracing its historical origin, exploring its wide-ranging connections across research areas, and highlighting some of the important results it has inspired.