An Algebraic Proof of the Polynomial Version of van der Waerden's Theorem

Javad Jafari; Mohammad Akbari Tootkaboni

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arXiv · 2508.06090

An Algebraic Proof of the Polynomial Version of van der Waerden's Theorem

Javad Jafari·Mohammad Akbari Tootkaboni

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Abstract

The polynomial version of van der Waerden's theorem, proved using dynamical systems by V. Bergelson and A. Leibman in 1996, \cite{Bergelson1996}, significantly highlighted the role of dynamical systems in addressing problems related to monochromatic configurations within algebraic structures. In this paper, by introducing symbolic polynomials, we aim to provide an alternative proof of the polynomial version of van der Waerden's theorem relying solely on Stone-\v{C}ech compactification of an infinite discrete semigroup.

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Javad Jafari, Mohammad Akbari Tootkaboni. 2025-08-08. An Algebraic Proof of the Polynomial Version of van der Waerden's Theorem. https://arxiv.org/abs/2508.06090

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