Bootstrapping partition regularity of linear systems. Issue 3 (9th August 2020) - Sanders, Tom
Abstract: Suppose that A is a k × d matrix of integers and write $\Re _A:{\mathbb N}→ {\mathbb N}\cup \{ ∞ \} $ 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 ℜ A ( 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 O A (1) term.