Modular-intersection graph Ramsey lower bound
= 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 $\binom{p^3}{p^2}\geq p^{p^2}$ vertices,
$$
R(p^{3p},p^{3p})\geq p^{p^2},
$$
a lower bound larger than every fixed power of $p^{3p}$ as $p\to\infty$.