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 F-p, where p equivalent to 1 (mod 4). We extend their idea to the finite field F-q, where q = p(2s+1) for a prime p equivalent to 1 (mod 4) and a non-negative integer s. We show the clique number of the Paley graph over Fp2 epsilon+1 is at most min (p(s)inverted right perpendicular root p/2inverted left perpendicular, root q/2 + p(s)+1/4 + root 2p/32p(s-1). (C) 2021 Elsevier Inc. All rights reserved.