  Let $q$ be an odd power of a prime $p$, and $S \subset \mathbb{F}_q^*$ such
that $S=-S$ and $S/S \neq \mathbb{F}_q^*$. We show that the clique number of
the Cayley graph $\operatorname{Cay}(\mathbb{F}_q^+,S)$ is at most
$\sqrt{|S/S|}+\sqrt{q/p}$, improving the best-known $\sqrt{q}$ upper bound for
many families of such graphs substantially. Such a new bound is strongest for
cyclotomic graphs and in particular, it implies the first nontrivial upper
bound on the clique number of all generalized Paley graphs of non-square order,
extending the work of Hanson and Pertidis. Moreover, our new bound is
asymptotically sharp for an infinite family of generalized Paley graphs, and we
further discover the first nontrivial family among them for which the clique
number can be exactly determined. We also obtain a new lower bound on the
number of directions determined by a large Cartesian product in the affine
Galois plane $AG(2,q)$, which is sharp for infinite families.
