Erdős–Rényi random graph and high probability

PDF p. 4: Section 1 random graph paragraph and footnote 4; PDF p. 30: Section 6, Erdős–Rényi random graph

For $n\in \mathbb{N}$ and $p\in [0,1]$, the Erdős–Rényi random graph $G(n,p)$ is a random simple graph on the labelled vertex set $[n]=\{1,\dots ,n\}$. A simple graph has an edge set consisting of unordered pairs of distinct vertices, so it has neither loops nor repeated edges. For each of the $\left(\genfrac{}{}{0px}{}{n}{2}\right)$ possible pairs, the indicator of its presence is $1$ with probability $p$ and $0$ with probability $1-p$; these indicators are mutually independent, meaning that every finite joint assignment has probability equal to the product of its individual probabilities.

For a sequence $p_n\in [0,1]$, let $E_n$ be an event, meaning a set of possible graphs on $[n]$, under the law of $G(n,p_n)$. The events $E_n$ hold with high probability if $\mathbb{P}(E_n)\to 1$ as $n\to \mathrm{\infty }$: for every $\epsilon >0$ there is an $N$ such that $\mathbb{P}(E_n)>1-\epsilon$ whenever $n\ge N$. The edge probability may therefore vary with $n$.

The finite product of Bernoulli edge spaces gives exactly the stated independent indicators, including at $p=0$ and $p=1$; its outcomes correspond uniquely to simple graphs on $[n]$. Thus the graph law is well defined by The Erdős-Rényi finite random graph G(n,p). The probability of an event is the sum of the weights of its graph outcomes. The high-probability convention is the ordinary limit of these event probabilities, even though the probability spaces vary with $n$, as specified in Asymptotic quantifiers and uniform parameter ranges.

The article uses these conventions in Section 1 and Section 6. Its citation records attribution, not a proof of the definition.