{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:KMGQKJKHBTPG4PKYKVMPPFQ77R","short_pith_number":"pith:KMGQKJKH","schema_version":"1.0","canonical_sha256":"530d0525470cde6e3d585558f7961ffc44841d0db9144bea8a7cb271e1874e9c","source":{"kind":"arxiv","id":"1010.6278","version":1},"attestation_state":"computed","paper":{"title":"Random graphs with few disjoint cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Colin McDiarmid, Valentas Kurauskas","submitted_at":"2010-10-29T17:17:42Z","abstract_excerpt":"The classical Erd\\H{o}s-P\\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G-B has no cycles. We show that, amongst all such graphs on vertex set {1,..,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class; concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the p"},"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":"1010.6278","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-29T17:17:42Z","cross_cats_sorted":[],"title_canon_sha256":"a9792cd14ab6b17a2da63643d6b1ab0f05ca8ee4cd84da20443262071bb10d15","abstract_canon_sha256":"32c6769235584369c15110d5665c36954cc5995bb9863c2fe00d27d954962b03"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:40.655918Z","signature_b64":"/iefginSC605Tl5DzM6b4Ep6YcELho4i6miED2dkEZslxqdQd+bG4FsDb5eK4/0TMqeuaCJoP/oUh6GZ6vBSAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"530d0525470cde6e3d585558f7961ffc44841d0db9144bea8a7cb271e1874e9c","last_reissued_at":"2026-05-18T03:43:40.655085Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:40.655085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random graphs with few disjoint cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Colin McDiarmid, Valentas Kurauskas","submitted_at":"2010-10-29T17:17:42Z","abstract_excerpt":"The classical Erd\\H{o}s-P\\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G-B has no cycles. We show that, amongst all such graphs on vertex set {1,..,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class; concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.6278","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1010.6278","created_at":"2026-05-18T03:43:40.655228+00:00"},{"alias_kind":"arxiv_version","alias_value":"1010.6278v1","created_at":"2026-05-18T03:43:40.655228+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.6278","created_at":"2026-05-18T03:43:40.655228+00:00"},{"alias_kind":"pith_short_12","alias_value":"KMGQKJKHBTPG","created_at":"2026-05-18T12:26:09.077623+00:00"},{"alias_kind":"pith_short_16","alias_value":"KMGQKJKHBTPG4PKY","created_at":"2026-05-18T12:26:09.077623+00:00"},{"alias_kind":"pith_short_8","alias_value":"KMGQKJKH","created_at":"2026-05-18T12:26:09.077623+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KMGQKJKHBTPG4PKYKVMPPFQ77R","json":"https://pith.science/pith/KMGQKJKHBTPG4PKYKVMPPFQ77R.json","graph_json":"https://pith.science/api/pith-number/KMGQKJKHBTPG4PKYKVMPPFQ77R/graph.json","events_json":"https://pith.science/api/pith-number/KMGQKJKHBTPG4PKYKVMPPFQ77R/events.json","paper":"https://pith.science/paper/KMGQKJKH"},"agent_actions":{"view_html":"https://pith.science/pith/KMGQKJKHBTPG4PKYKVMPPFQ77R","download_json":"https://pith.science/pith/KMGQKJKHBTPG4PKYKVMPPFQ77R.json","view_paper":"https://pith.science/paper/KMGQKJKH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1010.6278&json=true","fetch_graph":"https://pith.science/api/pith-number/KMGQKJKHBTPG4PKYKVMPPFQ77R/graph.json","fetch_events":"https://pith.science/api/pith-number/KMGQKJKHBTPG4PKYKVMPPFQ77R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KMGQKJKHBTPG4PKYKVMPPFQ77R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KMGQKJKHBTPG4PKYKVMPPFQ77R/action/storage_attestation","attest_author":"https://pith.science/pith/KMGQKJKHBTPG4PKYKVMPPFQ77R/action/author_attestation","sign_citation":"https://pith.science/pith/KMGQKJKHBTPG4PKYKVMPPFQ77R/action/citation_signature","submit_replication":"https://pith.science/pith/KMGQKJKHBTPG4PKYKVMPPFQ77R/action/replication_record"}},"created_at":"2026-05-18T03:43:40.655228+00:00","updated_at":"2026-05-18T03:43:40.655228+00:00"}