Uniform influence and sharp thresholds for graph and hypergraph properties

Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed $0\lt \varepsilon\lt 1/2$, every nontrivial increasing relabeling-invariant property crosses from probability ε to $1-\varepsilon$ within width $O((\log n)^{-2})$ for graphs and $O_r((\log n)^{-r/(r-1)})$ for r-uniform hypergraphs, r ≥ 3. The hypergraph influence bound also applies to nonmonotone properties.