  Suppose that $A$ is a $k \times d$ matrix of integers and write
$\mathfrak{R}_A:\mathbb{N} \rightarrow \mathbb{N}\cup \{ \infty\}$ for the
function taking $r$ to the largest $N$ such that there is an $r$-colouring
$\mathcal{C}$ of $[N]$ with $\bigcup_{C \in \mathcal{C}}{C^d}\cap \ker A
=\emptyset$. We show that if $\mathfrak{R}_A(r)<\infty$ for all $r \in
\mathbb{N}$ then $\mathfrak{R}_A(r) \leq \exp (\exp(r^{O_{A}(1)}))$ for all $r
\geq 2$.
  When the kernel of $A$ consists only of Brauer configurations -- that is
vectors of the form $(y,x,x+y,\dots,x+(d-2)y)$ -- the above has been proved by
Chapman and Prendiville with good bounds on the $O_A(1)$ term.
