Monotone graph property

A graph property is increasing and monotone when adding edges cannot destroy it.

A function $p^{\ast }(n)$ is a threshold when the property holds with probability tending to zero for $p/p^{\ast }\to 0$ and with probability tending to one for $p/p^{\ast }\to \infty$.

Assign an independent uniform label $U_e$ to every possible edge and include $e$ in $G(n,p)$ exactly when $U_e\le p$. This couples all edge probabilities so that $p\le q$ implies $G(n,p)\subseteq G(n,q)$.