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It is known that the number of directions formed by a Cartesian product $A\times B \subset AG(2,p)$ is at least $|A||B| - \min\{|A|,|B|\} + 2$, provided$p$ is prime and $|A||B|<p$. This implies the best known upper bound on theclique number of the Paley graph over $\mathbb{F}_p$. In this paper, we extendthis result to $AG(2,q)$, where $q$ is a prime power. We also give improvedupper bounds on the clique number of generalized Paley graphs over$\mathbb{F}_q$. In particular, for a cubic Paley graph, we improve the trivialupper bound $\sqrt{q}$ to $0.769\sqrt{q}+1$. In general, as an application ofour key result on the number of directions, for any positive function $h$ suchthat $h(x)=o(x)$ as $x \to \infty$, we improve the trivial upper bound$\sqrt{q}$ to $\sqrt{q}-h(p)$ for almost all non-squares $q$.

On the directions determined by Cartesian products and the clique number of generalized Paley graphs