{"id":"openai-math/A-uniform-influence-bound-for-hypergraph-properties","pdf":"https://github.com/openai/math/blob/main/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026","tags":["boolean-functions","hypergraphs","threshold-phenomena"],"score":0,"title":"A uniform influence bound for hypergraph properties","domain":"Combinatorics","family":"186","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-10-05","reactions":{},"discussion":[{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=together%20with%20the%20conjectured%20extension%20to%20uniform%20hypergraphs","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"together with the conjectured extension to uniform hypergraphs . Unique Games and 2-to-1 Games. My 2018 post on the 2-to-2 Games Theorem discussed progress toward Khot’s conjectures and the issue of perfect completeness. The release announces a proof of the Unique Games Conjecture and, separately,…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"}],"formalized":false,"family_title":"Uniform influence and sharp thresholds for graph and hypergraph properties"}