Some remarks on Alweiss's technique and monochromatic configuration of the form $\left\{ x,y,x+y,x\cdot y\right\} $ over Rationals (2212.13100v2)

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}.