On the Chromatic Number of the Erdős-Rényi Orthogonal Polarity Graph

For a prime power q, let ERq denote the Erdős-Rényi orthogonal polarity graph. We prove that if q is an even power of an odd prime, then χ(ERq) 6 2 √ q + O( √ q/ log q). This upper bound is best possible up to a constant factor of at most 2. If q is an odd power of an odd prime and satisfies some condition on irreducible polynomials, then we improve the best known upper bound for χ(ERq) substantially. We also show that for sufficiently large q, every ERq contains a subgraph that is not 3-chromatic and has at most 36 vertices.