A Canonical Polynomial Van der Waerden's Theorem

TL;DR

This paper proves a generalized polynomial version of Van der Waerden's theorem, showing that in any coloring of integers, certain polynomial-generated sets are guaranteed to be monochromatic or rainbow.

Contribution

It introduces a canonical polynomial Van der Waerden's theorem, extending classical results to polynomial configurations in integer colorings.

Findings

Existence of polynomial configurations that are monochromatic or rainbow in any coloring

Generalization of Van der Waerden's theorem to polynomial sequences

Applicable to any finite coloring of integers

Abstract

We prove a canonical polynomial Van der Waerden's Theorem. More precisely, we show the following. Let $\{p_1(x),\ldots,p_k(x)\}$ be a set of polynomials such that $p_i(x)\in \mathbb{Z}[x]$ and $p_i(0)=0$, for every $i\in \{1,\ldots,k\}$. Then, in any colouring of $\mathbb{Z}$, there exist $a,d\in \mathbb{Z}$ such that $\{a+p_1(d),\ldots,a+p_{k}(d)\}$ forms either a monochromatic or a rainbow set.