Uniform influence and sharp thresholds for graph and hypergraph properties

What OpenAI claims

Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed 0 < ε < 1/2, every nontrivial increasing relabeling-invariant property crosses from probability ε to 1-ε 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. (Lean)

Papers (2)

1. A uniform influence bound for hypergraph properties

   5 October 2026 · Preprint, not peer reviewed · PDF

   For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies Var _p(f) ≤ C_r I_p(f)/(log n)^r/(r-1). The constant depends only on r, and the bound holds uniformly for all 0 < p < 1 without a monotonicity assumption. For increasing properties, it gives the corresponding threshold-width bound with exponent r/(r-1), proving the hypergraph threshold-width conjecture of Friedgut and Kalai.

2. A Sharp Threshold Bound for Monotone Graph Properties

   25 September 2026 · Preprint, not peer reviewed · PDF

   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 < ε < 1/2, the edge probabilities at which its probability equals ε and 1-ε differ by at most C log(1/(2ε))/(log n)², for a universal constant C.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.