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A giant component in a sequence of random graphs on $n$ vertices has order proportional to $n$. Its existence can change at a threshold function for a monotone graph property. A giant component need not contain every vertex, and its uniqueness requires proof for the model in question.

For a monotone graph property $P=(P_n)$, a function $p^{\ast }(n)$ is a threshold function for a monotone graph property whenand