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Cooley, Do, Erde, and Missethan proved that for any fixed $\\alpha>0$, $G_{\\alpha n}(n,\\frac{1+\\epsilon}{n})$ with high probability contains a rainbow tree (a tree that does not repeat colors) which covers $(1\\pm O(\\epsilon))\\frac{\\alpha}{\\alpha+1}\\epsilon n$ vertices, and conjectur"},"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":"2308.14141","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-08-27T15:42:17Z","cross_cats_sorted":[],"title_canon_sha256":"766cbc603a06295b67cdec93eadf1d7dfdfbba33b9ba6e578efca0c8f4d7b9ae","abstract_canon_sha256":"427f008c8ffe6fa6d247e1780a513adea0839216179559fd7f0130f891ed59c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:45:22.264330Z","signature_b64":"yt9/Sb6KGhOWgQ8vWUFHh3+E4h3PRHhB+KrfZjYp8itNGLFtOh5aLHqmqaxm3g/5aGVNeHdRskeeAbuxS7/rCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70c90d533f40793fda2086f827e25fa9920fbfa259fedd679f6fff341e65b26e","last_reissued_at":"2026-07-05T06:45:22.263944Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:45:22.263944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Giant Rainbow Trees in Sparse Random Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Frieze, Tolson Bell","submitted_at":"2023-08-27T15:42:17Z","abstract_excerpt":"For any small constant $\\epsilon>0$, the Erd\\H{o}s-R\\'enyi random graph $G(n,\\frac{1+\\epsilon}{n})$ with high probability has a unique largest component which contains $(1\\pm O(\\epsilon))2\\epsilon n$ vertices. Let $G_c(n,p)$ be obtained by assigning each edge in $G(n,p)$ a color in $[c]$ independently and uniformly. Cooley, Do, Erde, and Missethan proved that for any fixed $\\alpha>0$, $G_{\\alpha n}(n,\\frac{1+\\epsilon}{n})$ with high probability contains a rainbow tree (a tree that does not repeat colors) which covers $(1\\pm O(\\epsilon))\\frac{\\alpha}{\\alpha+1}\\epsilon n$ vertices, and conjectur"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.14141","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2308.14141/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2308.14141","created_at":"2026-07-05T06:45:22.263997+00:00"},{"alias_kind":"arxiv_version","alias_value":"2308.14141v1","created_at":"2026-07-05T06:45:22.263997+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2308.14141","created_at":"2026-07-05T06:45:22.263997+00:00"},{"alias_kind":"pith_short_12","alias_value":"ODEQ2UZ7IB4T","created_at":"2026-07-05T06:45:22.263997+00:00"},{"alias_kind":"pith_short_16","alias_value":"ODEQ2UZ7IB4T7WRA","created_at":"2026-07-05T06:45:22.263997+00:00"},{"alias_kind":"pith_short_8","alias_value":"ODEQ2UZ7","created_at":"2026-07-05T06:45:22.263997+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/ODEQ2UZ7IB4T7WRAQ34CPYS7VG","json":"https://pith.science/pith/ODEQ2UZ7IB4T7WRAQ34CPYS7VG.json","graph_json":"https://pith.science/api/pith-number/ODEQ2UZ7IB4T7WRAQ34CPYS7VG/graph.json","events_json":"https://pith.science/api/pith-number/ODEQ2UZ7IB4T7WRAQ34CPYS7VG/events.json","paper":"https://pith.science/paper/ODEQ2UZ7"},"agent_actions":{"view_html":"https://pith.science/pith/ODEQ2UZ7IB4T7WRAQ34CPYS7VG","download_json":"https://pith.science/pith/ODEQ2UZ7IB4T7WRAQ34CPYS7VG.json","view_paper":"https://pith.science/paper/ODEQ2UZ7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2308.14141&json=true","fetch_graph":"https://pith.science/api/pith-number/ODEQ2UZ7IB4T7WRAQ34CPYS7VG/graph.json","fetch_events":"https://pith.science/api/pith-number/ODEQ2UZ7IB4T7WRAQ34CPYS7VG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ODEQ2UZ7IB4T7WRAQ34CPYS7VG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ODEQ2UZ7IB4T7WRAQ34CPYS7VG/action/storage_attestation","attest_author":"https://pith.science/pith/ODEQ2UZ7IB4T7WRAQ34CPYS7VG/action/author_attestation","sign_citation":"https://pith.science/pith/ODEQ2UZ7IB4T7WRAQ34CPYS7VG/action/citation_signature","submit_replication":"https://pith.science/pith/ODEQ2UZ7IB4T7WRAQ34CPYS7VG/action/replication_record"}},"created_at":"2026-07-05T06:45:22.263997+00:00","updated_at":"2026-07-05T06:45:22.263997+00:00"}