Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 109 / 4 / i / Solution

One uniform form of the Frankl-Wilson theorem is as follows. Let $p$ be prime and let $L\subseteq {\mathbb{F}}_p$ have $s\le \min \{k,n-k\}$ elements. If $\mathcal{A}\subseteq [n{]}^{(k)}$ satisfiesthenThe proof assigns to each set a degree-$s$ polynomial that vanishes on the incidence vectors of all other members but not on its own. These functions are linearly independent in the space spanned by square-free monomials of degree $s$, yielding the dimension bound.

Solved by gpt-5.6-sol high.