OpenAI · 186-02 · 2026-09-25 · Combinatorics
A Sharp Threshold Bound for Monotone Graph Properties
preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026
Family 186. The Friedgut–Kalai graph and hypergraph threshold conjectures.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed , every nontrivial increasing relabeling-invariant property crosses from probability to within width for graphs and for -uniform hypergraphs, . The hypergraph influence bound also applies to nonmonotone 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 , the edge probabilities at which its probability equals ε and differ by at most , for a universal constant C.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 186: Uniform influence and sharp thresholds for graph and hypergraph properties. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.