{"id":"openai-math/Talagrands-discrete-convexity-conjecture","pdf":"https://github.com/openai/math/blob/main/preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Talagrands-discrete-convexity-conjecture-September-23-2026","tags":["probability","combinatorics","discrete-convexity"],"score":0,"title":"Talagrand’s discrete-convexity conjecture","domain":"Combinatorics","family":"175","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"We prove Talagrand's discrete-convexity conjecture. There is a universal integer k such that, whenever an arbitrary family has Bernoulli product measure at least $`1-1/k`$, the sets not contained in a union of k members admit a cover of total cost at most 1/2 at the same density.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-23","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":"Talagrand’s expectation thresholds, discrete convexity, and graph decompositions"}