# Partition regularity and multiplicatively syndetic sets

## TL;DR

This paper links multiplicatively syndetic sets to the partition regularity of dilation invariant polynomial systems, providing new characterizations and bounds using adapted density increment strategies.

## Contribution

It establishes that a dilation invariant polynomial system is partition regular iff it appears in every multiplicatively syndetic set, and develops a syndetic density increment method.

## Key findings

- Partition regularity characterized by multiplicatively syndetic sets
- Bounds on Rado numbers for specific configurations derived
- Adaptation of Green-Tao and Chow-Lindqvist-Prendiville methods

## Abstract

We show how multiplicatively syndetic sets can be used in the study of partition regularity of dilation invariant systems of polynomial equations. In particular, we prove that a dilation invariant system of polynomial equations is partition regular if and only if it has a solution inside every multiplicatively syndetic set. We also adapt the methods of Green-Tao and Chow-Lindqvist-Prendiville to develop a syndetic version of Roth's density increment strategy. This argument is then used to obtain bounds on the Rado numbers of configurations of the form $\{x, d, x + d, x + 2d\}$.