On the independence number of graphs related to a polarity

We investigate the independence number of two graphs constructed from a polarity of PG(2, q). For the first graph under consideration, the Erdős-Rényi graph E, we provide an improvement on the known lower bounds on its independence number. In the second part of the paper, we consider the Erdős-Rényi hypergraph of triangles H _q. We determine the exact magnitude of the independence number of H _q, q even. This solves a problem posed by Mubayi and Williford [On the independence number of the ErdŐs-RÉnyi and projective norm graphs and a related hypergraph, J. Graph Theory, 56 (2007), pp. 113-127, Open Problem 3].