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 $p\to\infty$.

Modular-intersection graph Ramsey lower bound

For each $A\in\mathcal A$, define over $\mathbb F_p$
$$
P_A(x_1,\ldots,x_n)
=\prod_{e\in E}\left(\sum_{i\in A}x_i-e\right).
$$
Replace every power $x_i^r$ with $r\geq1$ by $x_i$; this does not change the values on <characteristic vectors of sets> and produces a multilinear polynomial of degree at most $m$.

At the characteristic vector of $B\in\mathcal A$,
$$
P_A(\mathbf1_B)=\prod_{e\in E}(|A\cap B|-e).
$$
This is zero when $A\ne B$, whereas
$$
P_A(\mathbf1_A)=\prod_{e\in E}(|A|-e)\ne0.
$$
The evaluation matrix is diagonal with nonzero diagonal, so the polynomials $P_A$ are <linearly independent vectors>[linearly independent]. The space of multilinear polynomials of degree at most $m$ has the monomial basis $\prod_{i\in S}x_i$ for $|S|\leq m$ and dimension $\sum_{i=0}^m\binom ni$. Hence the <modular intersection bound for a set family> gives
$$
\boxed{|\mathcal A|\leq\binom n0+\binom n1+\cdots+\binom nm}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 161 / 4 / i / Solution

Let $E\subseteq\mathbb F_p$ have size $m$, and let $\mathcal A\subseteq\mathcal P([n])$ satisfy $|A|\notin E$ and $|A\cap B|\in E$ for distinct $A,B\in\mathcal A$. Then
$$
|\mathcal A|\leq\sum_{i=0}^m\binom ni.
$$
The multilinearizations of
$$
P_A(x)=\prod_{e\in E}\left(\sum_{i\in A}x_i-e\right)
$$
are linearly independent functions on the characteristic vectors of the family and lie in the multilinear polynomial space of degree at most $m$.


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