A uniform influence bound for hypergraph properties
For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r -uniform hypergraphs on n vertices satisfies $\mathop{\mathrm{Var}}\nolimits _p(f)\le C_r I_p(f)/(\log n)^{r/(r-1)}$. The constant depends only on r , and the bound holds uniformly for all $0\lt p\lt 1$ without a monotonicity assumption. For increasing properties, it gives the corresponding threshold-width bound with exponent $r/(r-1)$, proving the hypergraph threshold-width conjecture of Friedgut and Kalai.
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
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