{"id":"openai-math/The-second-Kahn-Kalai-conjecture","pdf":"https://github.com/openai/math/blob/main/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026","tags":["random-graphs","combinatorics","thresholds"],"score":0,"title":"The second Kahn–Kalai conjecture","domain":"Combinatorics","family":"176","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"We prove the second Kahn–Kalai conjecture. For every finite simple graph H with h ≥ 1 edges and at most n vertices, the threshold for $`G(n,p)`$ to contain an ordinary copy of H is at most $`C p_{\\mathrm E}(n,H)(1+\\log_2 h)`$, where C is universal. Here $`p_{\\mathrm E}(n,H)`$ is the least density at which every subgraph of H has expected copy count at least one half.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-24","reactions":{},"discussion":[{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=reports%20proofs%20of%20the%20second%20Kahn%E2%80%93Kalai%20conjecture","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"reports proofs of the second Kahn–Kalai conjecture , which bounds graph-containment thresholds using subgraph expectations, and of Talagrand’s discrete convexity conjecture , both discussed in my account of Jinyoung Park’s ICM lecture . A companion result makes integral and fractional expectation…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"}],"formalized":true,"family_title":"The second Kahn–Kalai conjecture with an edge-count bound"}