Triangle-free induced subgraphs of polarity graphs

Abstract

Given a finite projective plane $Π$ and a polarity $θ$ of $Π$, the corresponding polarity graph is the graph whose vertices are the points of $Π$. Two distinct vertices $p$ and $p'$ are adjacent if $p$ is incident to $θ(p')$. Polarity graphs have been used in a variety of extremal problems, perhaps the most well-known being the Turán number of the cycle of length four. We investigate the problem of finding the maximum number of vertices in an induced triangle-free subgraph of a polarity graph. Mubayi and Williford showed that when $Π$ is the projective geometry $PG(2,q)$ and $θ$ is the orthogonal polarity, an induced triangle-free subgraph has at most $\frac{1}{2}q^2 + O(q^{3/2})$ vertices. We generalize this result to all polarity graphs, and provide some interesting computational results that are relevant to an unresolved conjecture of Mubayi and Williford.