Application of Hales-Jewett theorem near 0

TL;DR

This paper proves a near zero version of the polynomial van der Waerden theorem using combinatorics, addressing an unsolved problem related to arithmetic progressions in partitions.

Contribution

It introduces a novel proof of the near zero polynomial van der Waerden theorem, expanding the understanding of arithmetic progressions in combinatorics.

Findings

Proves the near 0 polynomial van der Waerden theorem

Uses combinatorial methods for the proof

Addresses an open problem in the field

Abstract

The famous van der Waerden theorem states that if partition N into finitely many cells then one of them will contain arbitrary length arithmetic progressions. It has a polynomial version also. In this article we will prove the near 0 version of the polynomial van der Waerden theorem which was unsolved, using combinatorics.