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

Theorem 1.1: for each $r\ge3$ there is $C_r$ with $\mathrm{Var}_p(f)\le C_rI_p(f)/(\log n)^{r/(r-1)}$ for every relabelling-invariant Boolean property of $r$-uniform hypergraphs, every $n\ge r$ and every $0<p<1$, without monotonicity. Corollary 4.1: for nontrivial increasing properties the width from $\varepsilon$ to $1-\varepsilon$ is $O_r(\log(1/\varepsilon)/(\log n)^{r/(r-1)})$. Not shown: explicit constants $C_r$ or sharpness examples for each $r$.

Theorem 1.1: for each $r\ge3$ there is $C_r$ with $\mathrm{Var}_p(f)\le C_rI_p(f)/(\log n)^{r/(r-1)}$ for every relabelling-invariant Boolean property of $r$-uniform hypergraphs, every $n\ge r$ and every $0<p<1$, without monotonicity. Corollary 4.1: for nontrivial increasing properties the width from $\varepsilon$ to $1-\varepsilon$ is $O_r(\log(1/\varepsilon)/(\log n)^{r/(r-1)})$. Not shown: explicit constants $C_r$ or sharpness examples for each $r$.