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For a prime power $q$, let $ER_q$ denote the Erd\H{o}s-R\'enyi orthogonalpolarity graph. We prove that if $q$ is an even power of an odd prime, then$\chi ( ER_{q}) \leq 2 \sqrt{q} + O ( \sqrt{q} / \log q)$. This upper bound isbest possible up to a constant factor of at most 2. If $q$ is an odd power ofan odd prime and satisfies some condition on irreducible polynomials, then weimprove the best known upper bound for $\chi(ER_{q})$ substantially. We alsoshow that for sufficiently large $q$, every $ER_q$ contains a subgraph that isnot 3-chromatic and has at most 36 vertices.

On the chromatic number of the Erd\H{o}s-R\'enyi orthogonal polarity graph