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 ${p_1(x),\ldots,p_k(x)}$ 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.