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Bootstrapping partition regularity of linear systems
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 63 / Issue 3 / August 2020
- Published online by Cambridge University Press:
- 09 March 2020, pp. 630-653
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- Article
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Suppose 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.
