Bootstrapping partition regularity of linear systems

Abstract

Suppose that $A$ is a $k\times d$ matrix of integers and write ${\mathfrak{R}}_A:\mathbb{N}\to \mathbb{N}\cup \{\infty \}$ for the function taking $r$ to the largest $N$ such that there is an $r$-colouring $\mathcal{C}$ of $[N]$ with ${\bigcup }_{C\in \mathcal{C}}{C}^d\cap \ker A=\varnothing$. We show that if ${\mathfrak{R}}_A(r)<\infty$ for all $r\in \mathbb{N}$ then ${\mathfrak{R}}_A(r)\le \exp (\exp (r^{O_A(1)}))$ for all $r\ge 2$. 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.

23 pp; corrections and an additional explanatory example from a referee

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