{"paper":{"title":"Perfect Matchings in Random Sparsifications of Dense Hypergraphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Han, Jingwen Zhao","submitted_at":"2025-07-15T14:28:14Z","abstract_excerpt":"Given \\(1\\le\\ell <k \\) and \\(\\delta\\geq 0\\), let \\(\\mathbf{PM}(k,\\ell,\\delta)\\) be the decision problem for the existence of perfect matchings in \\(n\\)-vertex \\(k\\)-uniform hypergraphs with minimum \\(\\ell\\)-degree at least \\(\\delta\\binom{n-\\ell}{k-\\ell}\\). For \\(k\\geq 3\\), \\(\\mathbf{PM}(k,\\ell,0)\\) was one of the first NP-complete problems identified by Karp. Keevash, Knox and Mycroft conjectured that \\(\\mathbf{PM}(k,\\ell,\\delta)\\) is in P for every \\(\\delta>1-(1-1/k)^{k-\\ell}\\) and this was recently verified by the work of Gan--Han, together with a very recent work of Fu et al.\n  In this pape"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.11359","kind":"arxiv","version":4},"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/2507.11359/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"}