Remark: Family 186. The Friedgut–Kalai graph and hypergraph threshold conjectures

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

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.