A Canonical Polynomial Van der Waerden's Theorem

Abstract

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

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