{"paper":{"title":"The Lov\\'{a}sz Local Lemma: Foundations and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The Lovász Local Lemma admits a proof based solely on unconditional probability inequalities.","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Igal Sason","submitted_at":"2026-03-07T14:55:00Z","abstract_excerpt":"The Lov\\'{a}sz Local Lemma (LLL) is a central tool in probabilistic combinatorics, providing a sufficient condition under which a finite collection of undesirable events with limited dependencies can be simultaneously avoided with positive probability. This paper offers a self-contained expository treatment of the lemma and its strengthened versions, emphasizing mathematical foundations, conceptual clarity, and applications. We begin with a pedagogically motivated proof of the LLL based entirely on unconditional probability inequalities. Particular attention is given to the symmetric form of t"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The paper presents a pedagogically motivated reformulation of the proof of the Lovász Local Lemma based solely on unconditional probability inequalities.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the reformulation using only unconditional probability inequalities is meaningfully simpler or more accessible than standard presentations in the literature.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An expository review presenting a pedagogically reformulated proof of the Lovász Local Lemma using unconditional inequalities, plus revisited applications and algorithmic perspectives.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Lovász Local Lemma admits a proof based solely on unconditional probability inequalities.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3abe45727fe73bb1b79794ce682980c139f53bc23aa10bcfc8015eb4bea82ccf"},"source":{"id":"2603.07245","kind":"arxiv","version":5},"verdict":{"id":"4d7f496d-7dd0-4348-94e2-83dc28b8e9b7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T14:25:52.389414Z","strongest_claim":"The paper presents a pedagogically motivated reformulation of the proof of the Lovász Local Lemma based solely on unconditional probability inequalities.","one_line_summary":"An expository review presenting a pedagogically reformulated proof of the Lovász Local Lemma using unconditional inequalities, plus revisited applications and algorithmic perspectives.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the reformulation using only unconditional probability inequalities is meaningfully simpler or more accessible than standard presentations in the literature.","pith_extraction_headline":"The Lovász Local Lemma admits a proof based solely on unconditional probability inequalities."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.07245/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"}