Bootstrapping partition regularity of linear systems

Abstract

AbstractSuppose that A is a k × d matrix of integers and write $\Re _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 =\emptyset $. We show that if ℜA(r) < ∞ for all $r\in {\mathbb N}$ then $\mathfrak {R}_A(r) \leqslant \exp (\exp (r^{O_{A}(1)}))$ for all r ⩾ 2. When the kernel of A consists only of Brauer configurations – that is, vectors of the form (y, x, x + y, …, x + (d − 2)y) – the above statement has been proved by Chapman and Prendiville with good bounds on the OA(1) term.