A Sharp Threshold Bound for Monotone Graph Properties

We prove the Friedgut–Kalai sharp-threshold conjecture. For every integer n ≥ 2, every nontrivial increasing family of graphs on n vertices invariant under all vertex permutations, and every $0<\epsilon <1/2$, the edge probabilities at which its probability equals ε and $1-\epsilon$ differ by at most $C\log (1/(2\epsilon ))/(\log n{)}^2$, for a universal constant C.