Paley Graphs and S\'ark\"ozy's Theorem In Function Fields

S'ark"ozy’s theorem states that dense sets of integers must contain two elements whose difference is a $k^{th}$ power. Following the polynomial method breakthrough of Croot, Lev, and Pach, Green proved a strong quantitative version of this result for ${\mathbb{F}}_q[T]$. In this paper we provide a lower bound for S'{a}rk"{o}zy’s theorem in function fields by adapting Ruzsa’s construction for the analogous problem in $\mathbb{Z}$. We construct a set $A$ of polynomials of degree $<n$ such that $A$ does not contain a $k^{th}$ power difference with $\mathrm{\mid }A\mathrm{\mid }=q^{n-n\mathrm{/}2k}$. Additionally, we prove a handful of results concerning the independence number of generalized Paley Graphs, including a generalization of a claim of Ruzsa, which helps with understanding the limit of the method.