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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 12F / iii / Solution

With

p = c lo g n / n

, the inequality

1 - p \leq e^{- p}

gives

E (N) = n (1 - p)^{n - 1} \leq n e^{- p (n - 1)} = n^{1 - c (n - 1) / n} ⟶ 0

when

c > 1

. The Markov inequality now gives

P (N > 0) \leq E (N) ⟶ 0.

Consequently the upper side of the isolated-vertex threshold in the Erdős-Rényi model is

P (N = 0) ⟶ 1 .

Solved by gpt-5.6-sol high.

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