for the clique numbers of strongly regular graphs in terms of their parameters. Our bounds improve on the Delsarte bound infinitely often.

Let $q = p^k$ be a power of a prime $p$ congruent to $1$ mod $4$. A **Paley graph** has vertex set equal to the finite field $\mathbb F_q$, and two vertices $a$ and $b$ are adjacent if and only if $a-b$ is a nonzero square. For a Paley graph $\Gamma$ on $q$ vertices with $k$ even, Blokhuis [@Blok84] showed that $\omega(\Gamma) = \sqrt{q}$; this corresponds to equality in the Delsarte bound. Bachoc et al. [@Bach:Paley13] recently considered the case when $\Gamma$ is a Paley graph on $q$ vertices with $k$ odd and, for certain such $q$, showed that $\omega(\Gamma) \leqslant \lfloor \sqrt{q} - 1 \rfloor$. This corresponds to an improvement to the Delsarte bound for these Paley graphs.

Here, working much more generally, given a strongly regular graph $\Gamma$ with parameters $(v,k,\lambda,\mu)$, we provide inequalities in terms of the parameters of $\Gamma$ that, when satisfied, guarantee that the clique number of $\Gamma$ is strictly less than the Delsarte bound. We show that these inequalities are satisfied by infinitely many feasible parameters tuples for strongly regular graphs and, 