On the independence number of the Erdős‐Rényi and projective norm graphs and a related hypergraph

The Erdős‐Rényi and Projective Norm graphs are algebraically defined graphs that have proved useful in supplying constructions in extremal graph theory and Ramsey theory. Their eigenvalues have been computed and this yields an upper bound on their independence number. Here we show that in many cases, this upper bound is sharp in the order of magnitude. Our result for the Erdős‐Rényi graph has the following reformulation: the maximum size of a family of mutually non‐orthogonal lines in a vector space of dimension three over the finite field of order q is of order q 3/2 . We also prove that every subset of vertices of size greater than q 2/2 + q 3/2 + O ( q ) in the Erdős‐Rényi graph contains a triangle. This shows that an old construction of Parsons is asymptotically sharp. Several related results and open problems are provided.