 Modular-intersection graph Ramsey lower bound

Let $p$ be prime and join two $p^2$-subsets of a $p^3$-element set when their intersection size is divisible by $p$. The modular intersection bound for a set family bounds both its independence and clique numbers by a quantity strictly below $p^{3p}$. Since the graph has $\left(\genfrac{}{}{0ex}{}{p^3}{p^2}\right)\ge p^{p^2}$ vertices,a lower bound larger than every fixed power of $p^{3p}$ as $p\to \infty$.