{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:QKAUJLGNZ57JD5RBXCXNCEJYQA","short_pith_number":"pith:QKAUJLGN","schema_version":"1.0","canonical_sha256":"828144accdcf7e91f621b8aed11138802f2a93fe3107a3d3d1968eb88708cb3d","source":{"kind":"arxiv","id":"1504.04392","version":3},"attestation_state":"computed","paper":{"title":"A Note on Weighted Rooted Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander York, Talon Ward, Zi-Xia Song","submitted_at":"2015-04-16T21:37:36Z","abstract_excerpt":"Let $T$ be a tree rooted at $r$. Two vertices of $T$ are related if one is a descendant of the other; otherwise, they are unrelated. Two subsets $A$ and $B$ of $V(T)$ are unrelated if, for any $a\\in A$ and $b\\in B$, $a$ and $b$ are unrelated. Let $\\omega$ be a nonnegative weight function defined on $V(T)$ with $\\sum_{v\\in V(T)}\\omega(v)=1$. In this note, we prove that either there is an $(r, u)$-path $P$ with $\\sum_{v\\in V(P)}\\omega(v)\\ge \\frac13$ for some $u\\in V(T)$, or there exist unrelated sets $A, B\\subseteq V(T)$ such that $\\sum_{a\\in A }\\omega(a)\\ge \\frac13$ and $\\sum_{b\\in B }\\omega(b)"},"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":"1504.04392","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-16T21:37:36Z","cross_cats_sorted":[],"title_canon_sha256":"3fa7ab34a340d826eec75f7a32c51b36eac1717d8fc4af217fd603530572ac85","abstract_canon_sha256":"92e49162515f24c272226396541aaa3bc435722bd86a26f80905e31aaef4ae5a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:13.335266Z","signature_b64":"hISGGEhwPY6mI2rMBkfw5MP3Z1/kEi4zD1gue0a4V1/oVm70pLC2cjmfd0XJ1jolW0j+Mxhv6+b10XoQeaDqAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"828144accdcf7e91f621b8aed11138802f2a93fe3107a3d3d1968eb88708cb3d","last_reissued_at":"2026-05-18T01:37:13.334639Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:13.334639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Note on Weighted Rooted Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander York, Talon Ward, Zi-Xia Song","submitted_at":"2015-04-16T21:37:36Z","abstract_excerpt":"Let $T$ be a tree rooted at $r$. Two vertices of $T$ are related if one is a descendant of the other; otherwise, they are unrelated. Two subsets $A$ and $B$ of $V(T)$ are unrelated if, for any $a\\in A$ and $b\\in B$, $a$ and $b$ are unrelated. Let $\\omega$ be a nonnegative weight function defined on $V(T)$ with $\\sum_{v\\in V(T)}\\omega(v)=1$. In this note, we prove that either there is an $(r, u)$-path $P$ with $\\sum_{v\\in V(P)}\\omega(v)\\ge \\frac13$ for some $u\\in V(T)$, or there exist unrelated sets $A, B\\subseteq V(T)$ such that $\\sum_{a\\in A }\\omega(a)\\ge \\frac13$ and $\\sum_{b\\in B }\\omega(b)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04392","kind":"arxiv","version":3},"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":"1504.04392","created_at":"2026-05-18T01:37:13.334755+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.04392v3","created_at":"2026-05-18T01:37:13.334755+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04392","created_at":"2026-05-18T01:37:13.334755+00:00"},{"alias_kind":"pith_short_12","alias_value":"QKAUJLGNZ57J","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"QKAUJLGNZ57JD5RB","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"QKAUJLGN","created_at":"2026-05-18T12:29:37.295048+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/QKAUJLGNZ57JD5RBXCXNCEJYQA","json":"https://pith.science/pith/QKAUJLGNZ57JD5RBXCXNCEJYQA.json","graph_json":"https://pith.science/api/pith-number/QKAUJLGNZ57JD5RBXCXNCEJYQA/graph.json","events_json":"https://pith.science/api/pith-number/QKAUJLGNZ57JD5RBXCXNCEJYQA/events.json","paper":"https://pith.science/paper/QKAUJLGN"},"agent_actions":{"view_html":"https://pith.science/pith/QKAUJLGNZ57JD5RBXCXNCEJYQA","download_json":"https://pith.science/pith/QKAUJLGNZ57JD5RBXCXNCEJYQA.json","view_paper":"https://pith.science/paper/QKAUJLGN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.04392&json=true","fetch_graph":"https://pith.science/api/pith-number/QKAUJLGNZ57JD5RBXCXNCEJYQA/graph.json","fetch_events":"https://pith.science/api/pith-number/QKAUJLGNZ57JD5RBXCXNCEJYQA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QKAUJLGNZ57JD5RBXCXNCEJYQA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QKAUJLGNZ57JD5RBXCXNCEJYQA/action/storage_attestation","attest_author":"https://pith.science/pith/QKAUJLGNZ57JD5RBXCXNCEJYQA/action/author_attestation","sign_citation":"https://pith.science/pith/QKAUJLGNZ57JD5RBXCXNCEJYQA/action/citation_signature","submit_replication":"https://pith.science/pith/QKAUJLGNZ57JD5RBXCXNCEJYQA/action/replication_record"}},"created_at":"2026-05-18T01:37:13.334755+00:00","updated_at":"2026-05-18T01:37:13.334755+00:00"}