Independent Sets in Polarity Graphs

Given a projective plane Σ and a polarity θ of Σ, the corresponding polarity graph is the graph whose vertices are the points of Σ, and two distinct points p1 and p2 are adjacent if p1 is incident to p θ 2 in Σ. A well-known example of a polarity graph is the Erdős-Rényi orthogonal polarity graph ERq, 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 ERq, the eigenvalue method gives the correct upper bound in order of magnitude. We prove that this is also true for 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 Ω(q3/2). Some related results are also obtained.