On the clique number of Paley graphs of prime power order

Finding a reasonably good upper bound for the clique number of Paley graphs is an open problem in additive combinatorics. A recent breakthrough by Hanson and Petridis using Stepanov’s method gives an improved upper bound on Paley graphs defined on a prime field ${\mathbb{F}}_p$, where $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$. We extend their idea to the finite field ${\mathbb{F}}_q$, where $q=p^{2s+1}$ for a prime $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and a non-negative integer $s$. We show the clique number of the Paley graph over ${\mathbb{F}}_{p^{2s+1}}$ is at most $\min \bigg(p^s \bigg\lceil \sqrt{\frac{p}{2}} \bigg\rceil, \sqrt{\frac{q}{2}}+\frac{p^s+1}{4}+\frac{\sqrt{2p}}{32}p^{s-1}\bigg)$.