A clique-free pseudorandom subgraph of the pseudo polarity graph

TL;DR

This paper introduces a new family of $K_k$-free pseudorandom graphs with specific edge density, extending previous constructions to include cases where the finite field has even characteristic, thus broadening the applicability of such graphs.

Contribution

The authors present the first construction of $K_k$-free pseudorandom graphs over finite fields with even characteristic, matching the edge density of previous odd-characteristic constructions.

Findings

Constructed $K_k$-free pseudorandom graphs with edge density $ heta(n^{-1/(k-1)})$

Extended polarity graph methods to finite fields with even characteristic

Matched the edge density of prior odd-characteristic constructions

Abstract

We provide a new family of $K_k$-free pseudorandom graphs with edge density $\Theta(n^{-1/(k-1)})$, matching a recent construction due to Bishnoi, Ihringer and Pepe. As in the former result, the idea is to use large subgraphs of polarity graphs, which are defined over a finite field $\mathbb{F}_q$. While their construction required $q$ to be odd, we will give the first construction with $q$ even.