 Edge-exposure martingale

Expose the independent edge indicators of a binomial random graph in a fixed order, and set $M_i=\mathbb{E}[Z\mid \text{the first }i\text{ indicators}]$ for an integrable statistic $Z$. This is a martingale from $\mathbb{E}Z$ to $Z$. If changing one edge changes $Z$ by at most $c$, coupling the unexposed indicators gives $|M_i-M_{i-1}|\le c$, permitting the Azuma-Hoeffding inequality.