Let $\mathcal A$ be an <independent set> in the graph. Every member has size $p^2\equiv0\pmod p$, while for distinct $A,B\in\mathcal A$, independence means $|A\cap B|$ is nonzero modulo $p$. Apply part i with
$$
E=\mathbb F_p\setminus\{0\},
\qquad m=p-1.
$$
This gives
$$
\alpha(G)\leq\sum_{i=0}^{p-1}\binom{p^3}{i}.
$$
Moreover,
$$
\sum_{i=0}^{p-1}\binom{p^3}{i}
\leq p(p^3)^{p-1}=p^{3p-2}<p^{3p}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-161/4/ii/solution