Uniform random graph process (source code)

= Uniform random graph process
{title2=$G_{n,m}$}

Start with the empty <graph> and reveal the <edges> of the <complete graph> in a uniformly random order. After $m$ steps its edge set is a uniform $m$-element subset. Independent uniform edge labels couple this process with all <binomial random graphs>: $G(n,p)=G_{n,M(p)}$, where $M(p)$ has a <binomial distribution> with parameters $\binom n2,p$. For a <monotone graph property>, concentration of $M(p)$ transfers a threshold estimate between edge probability and edge count.