{"paper":{"title":"A Proof of the Kahn-Kalai Conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","math.PR"],"primary_cat":"math.CO","authors_text":"Huy Tuan Pham, Jinyoung Park","submitted_at":"2022-03-31T17:32:03Z","abstract_excerpt":"Proving the ``expectation-threshold'' conjecture of Kahn and Kalai, we show that for any increasing property $\\mathcal{F}$ on a finite set $X$, $$p_c(\\mathcal{F})=O(q(\\mathcal{F})\\log \\ell(\\mathcal{F})),$$ where $p_c(\\mathcal{F})$ and $q(\\mathcal{F})$ are the threshold and ``expectation threshold'' of $\\mathcal{F}$, and $\\ell(\\mathcal{F})$ is the maximum of $2$ and the maximum size of a minimal member of $\\mathcal{F}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2203.17207","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2203.17207/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}