Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 186-02. A Sharp Threshold Bound for Monotone Graph Properties
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OpenAI@2026-10-07#rem:oai-p-186-02
Authors: OpenAI. Family 186.
Title: A Sharp Threshold Bound for Monotone Graph Properties.
Abstract: 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 $0{}lt {}varepsilon{}lt 1/2$, the edge probabilities at which its probability equals ε and $1-{}varepsilon$ differ by at most $C{}log(1/(2{}varepsilon))/({}log n){}2$, for a universal constant C.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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