The Friedgut-Kalai sharp-threshold conjecture: threshold width O(1/log^2 n) for monotone graph properties

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Comparator statement: SharpThreshold.lean evidence for this event

Friedgut-Kalai, Every monotone graph property has a sharp threshold (1996) evidence for this event

The Friedgut-Kalai threshold-width conjecture for r-uniform hypergraph properties evidence for this event

The Friedgut-Kalai sharp-threshold conjecture: threshold width O(1/log^2 n) for monotone graph properties parent of this event

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Lean proof module (OAI.Problem313.sharp_threshold_width) evidence for this event

Theorem 1.1: for every $n\ge2$, nontrivial increasing vertex-symmetric graph property and $0<\varepsilon<1/2$, $p_{1-\varepsilon}-p_\varepsilon\le2^{19}\log(1/(2\varepsilon))/(\log n)^2$. It follows from Theorem 1.2, $\mathrm{Var}_p(f)\le2^{17}I_p(f)/(\log n)^2$ for every vertex-symmetric Boolean $f$ and every $0<p<1$, without monotonicity, which removes the $(\log\log n)^2$ loss of Kelman-Kindler-Lifshitz-Minzer-Safra at $p=1/2$. The order is optimal for fixed $\varepsilon$ (clique example). Not shown: the best constant, or where thresholds lie (a different question). parent of this event

VibeMathed record: The Friedgut-Kalai sharp-threshold conjecture: threshold width O(1/log^2 n) for monotone graph properties evidence for this event

Companion: A Sharp Threshold Bound for Monotone Graph Properties evidence for this event