Some remarks on Alweiss's technique and monochromatic configuration of the form $\left\{ x,y,x+y,x\cdot y\right\} $ over Rationals

TL;DR

This paper investigates ultrafilter techniques to analyze monochromatic configurations of the form {x, y, x+y, x·y} over rationals, extending to configurations with powers, building on recent combinatorial and ultrafilter methods.

Contribution

It demonstrates that ultrafilters contain these configurations and generalizes results to include sets with powers, advancing the understanding of partition regularity over rationals.

Findings

Ultrafilters contain configurations of the form {x, y, x+y, x·y}.

Sets contain configurations of the form {x, y, x+y, x·y^n} for any n.

Extends previous results on monochromatic configurations over rationals.

Abstract

In this article, we will explore a recent method of Alweiss \cite{key-1} using ultrafilter technique to study monochromatic partition regular structure of the form $\left\{ x,y,x+y,x\cdot y\right\} $ over rationals, which is recently proved by Bowen, and Sabok in \cite{key-17}. Our methods explore that each member of combinatorially rich ultrafilters contains these types of configurations. Besides this, we will also prove that for any $n\in\mathbb{N},$ these sets will contain configuration of the form $\left\{ x,y,x+y,x\cdot y^{n}\right\}$, partially proved by Xiao in \cite{key-25}.