A clique-free pseudorandom subgraph of the pseudo polarity graph

We provide a new family of K_k-free pseudorandom graphs with edge density Θ(n^−1/(k−1)), matching a recent construction due to Bishnoi, Ihringer and Pepe [2]. As in the former result, the idea is to use large subgraphs of polarity graphs, which are defined over a finite field F_q. While their construction required q to be odd, we will give the first construction with q a power of 2.