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

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).

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).