A Canonical Polynomial Van der Waerdens Theorem

We prove a canonical polynomial Van der Waerdens 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 $iin {1,ldots,k}$. Then, in any colouring of $mathbb{Z}$, there exist $a,din mathbb{Z}$ such that ${a+p_1(d),ldots,a+p_{k}(d)}$ forms either a monochromatic or a rainbow set.