Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 109 / 4 / Solution

The Frankl-Wilson theorem says that if $p$ is prime, $L\subseteq {\mathbb{F}}_p$ has $s\le \min \{r,n-r\}$ elements, and $\mathcal{F}\subseteq [n{]}^{(r)}$ satisfiesthen $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{n}{s}\right)$.

For each $A\in \mathcal{F}$, form the multilinearization on the Boolean cube ofAt the characteristic vector of a set $1_B$, this polynomial vanishes for $B\ne A$ and is nonzero for $B=A$. Hence the restricted functions $P_A$ are linearly independent. On the $r$-slice, every square-free monomial of degree below $s$ can be raised to degree $s$ using the relation ${\sum }_i{x}_i=r$, so the degree-at-most-$s$ function space is spanned by the $\left(\genfrac{}{}{0ex}{}{n}{s}\right)$ square-free degree-$s$ monomials. Linear independence gives the theorem.