{"paper":{"title":"The First Time KE is Broken up","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adi Jarden","submitted_at":"2016-03-22T17:47:49Z","abstract_excerpt":"A relevant collection is a collection, $F$, of sets, such that each set in $F$ has the same cardinality, $\\alpha(F)$. A Konig Egervary (KE) collection is a relevant collection $F$, that satisfies $|\\bigcup F|+|\\bigcap F|=2\\alpha(F)$. An hke (hereditary KE) collection is a relevant collection such that all of his non-empty subsets are KE collections. In \\cite{jlm} and \\cite{dam}, Jarden, Levit and Mandrescu presented results concerning graphs, that give the motivation for the study of hke collections. In \\cite{hke}, Jarden characterize hke collections.\n  Let $\\Gamma$ be a relevant collection su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06887","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":""},"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"}