Abstract
Given a finite projective plane $\Pi$ and a polarity $\theta$ of $\Pi$, the
corresponding polarity graph is the graph whose vertices are the points of
$\Pi$. Two distinct vertices $p$ and $p'$ are adjacent if $p$ is incident to
$\theta (p')$. Polarity graphs have been used in a variety of extremal
problems, perhaps the most well-known being the Tur\'{a}n 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 $\Pi$ is the projective geometry $PG(2,q)$ and
$\theta$ 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.