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

Table of contents
- Solution ii

Solution

Let

A

be an independent set in the graph. Every member has size

p^{2} \equiv 0 (mod p)

, while for distinct

A, B \in A

, independence means

∣ A \cap B ∣

is nonzero modulo

p

. Apply part i with

E = F_{p} ∖ {0}, m = p - 1.

This gives

α (G) \leq \sum_{i = 0}^{p - 1} (i p ^{3}) .

Moreover,

\sum_{i = 0}^{p - 1} (i p ^{3}) \leq p (p^{3})^{p - 1} = p^{3 p - 2} < p^{3 p} .

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