{"id":"openai-math/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties","pdf":"https://github.com/openai/math/blob/main/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026","tags":["random-graphs","threshold-phenomena","combinatorics"],"score":0,"title":"A Sharp Threshold Bound for Monotone Graph Properties","domain":"Combinatorics","family":"186","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"We prove the Friedgut–Kalai sharp-threshold conjecture. For every integer n ≥ 2, every nontrivial increasing family of graphs on n vertices invariant under all vertex permutations, and every $`0\\lt \\varepsilon\\lt 1/2`$, the edge probabilities at which its probability equals ε and $`1-\\varepsilon`$ differ by at most $`C\\log(1/(2\\varepsilon))/(\\log n)^2`$, for a universal constant C .","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-25","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"}