On the independence number of graphs related to a polarity

TL;DR

This paper studies the independence number of graphs derived from a polarity in projective geometry, improving bounds and exactly determining values for specific cases, thus advancing understanding in combinatorial geometry.

Contribution

It provides improved lower bounds for the independence number of the Erd ext{o}s-Rényi graph and determines the exact independence number of the Erd ext{o}s-Rényi hypergraph of triangles for even q.

Findings

Improved lower bounds on the independence number of ER_q.

Exact independence number of al_q for even q.

Solved a problem posed by Mubayi and Williford.

Abstract

We investigate the independence number of two graphs constructed from a polarity of $\mathrm{PG}(2,q)$. For the first graph under consideration, the Erd\H{o}s-R\'enyi graph $ER_q$, we provide an improvement on the known lower bounds on its independence number. In the second part of the paper we consider the Erd\H{o}s-R\'enyi hypergraph of triangles $\mathcal{H}_q$. We determine the exact magnitude of the independence number of $\mathcal{H}_q$, $q$ even. This solves a problem posed by Mubayi and Williford.