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

Let $\mathcal{C}$ be a clique and choose the given prime $q$ with $p^2<q<2p^2$. For distinct $A,B\in \mathcal{C}$, the intersection size is a multiple of $p$ strictly below $p^2$. In ${\mathbb{F}}_q$, setThese are $p$ distinct residues because $q>p^2$, every off-diagonal intersection size lies in $E$, and the diagonal size $p^2$ does not. Part i, now over ${\mathbb{F}}_q$, yieldsThe subsets of a $p^3$-element set having size at most $p$ inject into its ordered $p$-tuples: list a nonempty subset increasingly and repeat its final element, while assigning the empty set one decreasing tuple not used in this way. The injection is not surjective, so