Capturing Forms in Dense Subsets of Finite Fields

An open problem of arithmetic Ramsey theory asks if given a finite $r$-colouring $c:\mathbb{N}\to\{1,...,r\}$ of the natural numbers, there exist $x,y\in \mathbb{N}$ such that $c(xy)=c(x+y)$ apart from the trivial solution $x=y=2$. More generally, one could replace $x+y$ with a binary linear form and $xy$ with a binary quadratic form. In this paper we examine the analogous problem in a finite field $\mathbb{F}_q$. Specifically, given a linear form $L$ and a quadratic from $Q$ in two variables, we provide estimates on the necessary size of $A\subset \mathbb{F}_q$ to guarantee that $L(x,y)$ and $Q(x,y)$ are elements of $A$ for some $x,y\in\mathbb{F}_q$.