 Friedgut-Kalai sharp threshold theorem

There is an absolute $C>0$ such that any nontrivial transitive increasing event on $n\ge 2$ independent bits satisfies ${\mu }_q(A)>1-\epsilon$ if ${\mu }_p(A)>\epsilon$, $0<\epsilon <1/2$, and the displayed bound holds with $p<q<1$. A uniform weighted Kahn-Kalai-Linial theorem gives one pivotal probability at least $c{\mu }_r(A)(1-{\mu }_r(A))\log (n)/n$. Transitivity makes the total influence at least $c{\mu }_r(A)(1-{\mu }_r(A))\log n$. The Margulis–Russo formula identifies this with the derivative of ${\mu }_r(A)$. Integration of the derivative of its log odds proves the threshold bound.