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

TL;DR

The paper provides an algebraic, purely combinatorial proof of the polynomial van der Waerden theorem using Stone–Čech compactification and adequate partial semigroups. It introduces symbolic polynomial spaces $V_k$ and $V(k,m)$, establishing a bridge to ordinary polynomials via surjective maps $P_x$ and $P_{x_1,...,x_k}$, and leverages ultrafilter/central-set techniques to obtain monochromatic polynomial configurations of the form $a+p_i(\sum_{t\in F}f(t))$. This approach yields both univariate and multivariable polynomial variants without ergodic theory, highlighting the role of Stone–Čech methods in Ramsey-theoretic polynomial configurations. The results provide a framework for purely combinatorial proofs of polynomial Ramsey phenomena and may extend to broader partial-semigroup contexts.

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-Čech compactification of an infinite discrete semigroup.

Theorems & Definitions (55)

• Theorem 1.1: Van der Waerden,vandw1927
• Theorem 1.2: Polynomial van der Waerden, Bergelson–Leibman, Bergelson1996
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Theorem 2.5
• proof
• Definition 3.1