  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.
