{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:QJWXEIJDYINFMVE2VAHO7IS7ZG","short_pith_number":"pith:QJWXEIJD","schema_version":"1.0","canonical_sha256":"826d722123c21a56549aa80eefa25fc9af87e80677d52659ad11943f17aafeb9","source":{"kind":"arxiv","id":"1103.2685","version":6},"attestation_state":"computed","paper":{"title":"Ramsey numbers for trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zhi-Hong Sun","submitted_at":"2011-03-14T15:07:18Z","abstract_excerpt":"For $n\\ge 5$ let $T_n'$ denote the unique tree on $n$ vertices with $\\Delta(T_n')=n-2$, and let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\\{v_0,v_1,\\ldots,$ $v_{n-1}\\}$ and $E=\\{v_0v_1,\\ldots,v_0v_{n-3},$ $v_{n-3}v_{n-2},v_{n-2}v_{n-1}\\}$. In this paper we evaluate the Ramsey numbers $r(G_m,T_n')$ and $r(G_m,T_n^*)$, where $G_m$ is a connected graph of order $m$. As examples, for $n\\ge 8$ we have $r(T_n',T_n^*)=r(T_n^*,T_n^*)=2n-5$, for $n>m\\ge 7$ we have $r(K_{1,m-1},T_n^*)=m+n-3$ or $m+n-4$ according as $m-1\\mid (n-3)$ or $m-1\\nmid (n-3)$, for $m\\ge 7$ and $n\\ge (m-3)^2+2$ we have $r"},"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":"1103.2685","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-14T15:07:18Z","cross_cats_sorted":[],"title_canon_sha256":"c937c3ea2ca1e718f71e9cd068066433d2a1866f4453ada3b29e77e05c470932","abstract_canon_sha256":"e6b57321915f15e79d3139fa297d4fea2f4d8cc7bb2ec501f1418d4eb3c3c4ac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:24.456143Z","signature_b64":"kwvjpIoAgmtt+arCiuUftlhJ2VWikxEzrAzTYk6zDKpD1YCs4BZAVIx58+3y0ujPsNgetI+04+KbsaKMusXNDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"826d722123c21a56549aa80eefa25fc9af87e80677d52659ad11943f17aafeb9","last_reissued_at":"2026-05-18T02:39:24.455649Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:24.455649Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ramsey numbers for trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zhi-Hong Sun","submitted_at":"2011-03-14T15:07:18Z","abstract_excerpt":"For $n\\ge 5$ let $T_n'$ denote the unique tree on $n$ vertices with $\\Delta(T_n')=n-2$, and let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\\{v_0,v_1,\\ldots,$ $v_{n-1}\\}$ and $E=\\{v_0v_1,\\ldots,v_0v_{n-3},$ $v_{n-3}v_{n-2},v_{n-2}v_{n-1}\\}$. In this paper we evaluate the Ramsey numbers $r(G_m,T_n')$ and $r(G_m,T_n^*)$, where $G_m$ is a connected graph of order $m$. As examples, for $n\\ge 8$ we have $r(T_n',T_n^*)=r(T_n^*,T_n^*)=2n-5$, for $n>m\\ge 7$ we have $r(K_{1,m-1},T_n^*)=m+n-3$ or $m+n-4$ according as $m-1\\mid (n-3)$ or $m-1\\nmid (n-3)$, for $m\\ge 7$ and $n\\ge (m-3)^2+2$ we have $r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.2685","kind":"arxiv","version":6},"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":"1103.2685","created_at":"2026-05-18T02:39:24.455722+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.2685v6","created_at":"2026-05-18T02:39:24.455722+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.2685","created_at":"2026-05-18T02:39:24.455722+00:00"},{"alias_kind":"pith_short_12","alias_value":"QJWXEIJDYINF","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_16","alias_value":"QJWXEIJDYINFMVE2","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_8","alias_value":"QJWXEIJD","created_at":"2026-05-18T12:26:39.201973+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/QJWXEIJDYINFMVE2VAHO7IS7ZG","json":"https://pith.science/pith/QJWXEIJDYINFMVE2VAHO7IS7ZG.json","graph_json":"https://pith.science/api/pith-number/QJWXEIJDYINFMVE2VAHO7IS7ZG/graph.json","events_json":"https://pith.science/api/pith-number/QJWXEIJDYINFMVE2VAHO7IS7ZG/events.json","paper":"https://pith.science/paper/QJWXEIJD"},"agent_actions":{"view_html":"https://pith.science/pith/QJWXEIJDYINFMVE2VAHO7IS7ZG","download_json":"https://pith.science/pith/QJWXEIJDYINFMVE2VAHO7IS7ZG.json","view_paper":"https://pith.science/paper/QJWXEIJD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.2685&json=true","fetch_graph":"https://pith.science/api/pith-number/QJWXEIJDYINFMVE2VAHO7IS7ZG/graph.json","fetch_events":"https://pith.science/api/pith-number/QJWXEIJDYINFMVE2VAHO7IS7ZG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QJWXEIJDYINFMVE2VAHO7IS7ZG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QJWXEIJDYINFMVE2VAHO7IS7ZG/action/storage_attestation","attest_author":"https://pith.science/pith/QJWXEIJDYINFMVE2VAHO7IS7ZG/action/author_attestation","sign_citation":"https://pith.science/pith/QJWXEIJDYINFMVE2VAHO7IS7ZG/action/citation_signature","submit_replication":"https://pith.science/pith/QJWXEIJDYINFMVE2VAHO7IS7ZG/action/replication_record"}},"created_at":"2026-05-18T02:39:24.455722+00:00","updated_at":"2026-05-18T02:39:24.455722+00:00"}