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In this paper we describe the tournaments which do not have $U_5$ as a subtournament. Specifically, we show that if a tournament $G$ is \"prime\"---that is, if there is no subset $X \\subseteq V(G)$, $1 < |X| < |V(G)|$, such that for all $v \\in V(G) \\backslash X$, either $v \\rightarrow x$ for all $x \\in X$ or $x \\rightarrow v$ for all $x \\in X$---then $G$ is $U_5$-free if and only if either $G$ is a specific tournament $T_n$","authors_text":"Gaku Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-02T04:37:19Z","title":"Structure theorem for U5-free tournaments"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.0398","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f9666ecba810dadf935a419bc30bf794396c3ea3d0e5d18dde4bdc057bacec36","target":"record","created_at":"2026-05-18T02:49:03Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3cd51d39ff7320c6e277bfbe4c80d266244863ae054aabc20b8e864756f0b988","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-02T04:37:19Z","title_canon_sha256":"22dae2176012796696dcaafe62bd87666dea52966d339b4aa0318bf2f537bfd7"},"schema_version":"1.0","source":{"id":"1208.0398","kind":"arxiv","version":2}},"canonical_sha256":"ac609d3cb7c08acaab9f2e7001484322bca9d4a0190ffb8fd606245d77081de0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac609d3cb7c08acaab9f2e7001484322bca9d4a0190ffb8fd606245d77081de0","first_computed_at":"2026-05-18T02:49:03.445692Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:49:03.445692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MP73aUGiJ8oT8zQav/a4MEkF4hVvUo7lfmTCSi//ek3x+M/DakA33P2xAIf5ZftWQRaB/x3q55yySjixIdcvDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:49:03.446134Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.0398","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f9666ecba810dadf935a419bc30bf794396c3ea3d0e5d18dde4bdc057bacec36","sha256:97bbc45c101f77eea48e07b25201c7247c4ffda9921c6dea3c462c204db85be8"],"state_sha256":"4bb599005ebef5cbe88a4fb0f654c320935ffe7671a9f6d5bc4cd446b410db06"}