# Bootstrapping partition regularity of linear systems

## TL;DR

This paper establishes an upper bound on the size of the largest set avoiding certain linear configurations under any coloring, showing a double exponential growth bound when the system's regularity function is finite.

## Contribution

It provides a double exponential upper bound on the partition regularity function for linear systems, extending previous results to a broader class of configurations.

## Key findings

- If the regularity function is finite for all r, then it is bounded by a double exponential in r.
- The result applies to systems where the kernel consists of Brauer configurations.
- The paper generalizes known bounds for specific configurations to more general linear systems.

## Abstract

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.