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The conjecture has also been proven for $r = 3$ by Aharoni using topological methods, but the proof does not give information on the extremal $3$-uniform hypergraphs. Our goal in this paper is to characterize those hypergraphs which are tight for Aharoni's Theorem.\n  Our proof of this characterization is also based on topological m"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1401.0171","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-31T16:33:15Z","cross_cats_sorted":[],"title_canon_sha256":"d5d1b4c4834a255227fc8ed13186bfc3b8f1d03c6f68adb19a9186fb0220344c","abstract_canon_sha256":"8103af27d59fac035447cc9153daa971c6c468550abdcf4964cdf2614f90ae44"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:18.877455Z","signature_b64":"9Ox7geBXHeUDvbdlXOAb+bApgHYQD39Jk4nDce2dM7rI6k9rLhZrFZ6OjfB9wYiTeUYue7CAPEFSamy35+J8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41cb57393940955121f45bb546c456a99d0a43c560573471dbf366df456a8677","last_reissued_at":"2026-05-18T01:12:18.877109Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:18.877109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extremal Hypergraphs for Ryser's Conjecture: Home-Base Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lothar Narins, Penny Haxell, Tibor Szab\\'o","submitted_at":"2013-12-31T16:33:15Z","abstract_excerpt":"Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph has a vertex cover of size at most $r - 1$ times the size of the largest matching. 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