Combinatorics · Oct 6, 2026

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.

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