Independent Sets in Polarity Graphs

Abstract

Given a projective plane $\Sigma$ and a polarity $\theta$ of $\Sigma$, the corresponding polarity graph is the graph whose vertices are the points of $\Sigma$, and two distinct points $p_1$ and $p_2$ are adjacent if $p_1$ is incident to $p_2^{ \theta}$ in $\Sigma$. A well-known example of a polarity graph is the Erdos--Renyi orthogonal polarity graph $ER_q$, which appears frequently in a variety of extremal problems. Eigenvalue methods provide an upper bound on the independence number of any polarity graph. Mubayi and Williford showed that in the case of $ER_q$, the eigenvalue method gives the correct upper bound in order of magnitude. We prove that this is also true for certain other families of polarity graphs. This includes a family of polarity graphs for which the polarity is neither orthogonal nor unitary. We conjecture that any polarity graph of a projective plane of order $q$ has an independent set of size $\Omega (q^{3/2})$. Some related results are also obtained.