# On the Ramsey number of the Brauer configuration

## TL;DR

This paper establishes new upper bounds on the Ramsey numbers related to Brauer's theorem and quadratic equations, improving understanding of color-structured progressions and nonlinear configurations.

## Contribution

It provides the first double exponential bound for Brauer's theorem and refines bounds for three-term progressions and quadratic equations using advanced combinatorial techniques.

## Key findings

- Double exponential bound for Brauer's theorem established.
- Refined bounds for three-term progressions in three colors.
- Bounds for Ramsey numbers of certain quadratic equations derived.

## Abstract

We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions with the same colour as their common difference. Such a result has been obtained independently and in much greater generality by Sanders. Using Gowers' local inverse theorem, our bound is quintuple exponential in the length of the progression. We refine this bound in the colour aspect for three-term progressions, and combine our arguments with an insight of Lefmann to obtain analogous bounds for the Ramsey numbers of certain nonlinear quadratic equations.