Paley graphs and sárközy’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 Croot, Lev, and Pach, Green proved strong quantitative version this result for $\mathbb{F}_{q}[T]$. In paper we provide lower bound S\'{a}rk\"{o}zy's in function fields by adapting Ruzsa's construction analogous problem $\mathbb{Z}$. We construct set $A$ polynomials degree $<n$ such does not power with $|A|=q^{n-n/2k}$. Additionally, prove handful results concerning independence number generalized Paley Graphs, including generalization claim Ruzsa, which helps understanding limit method.