{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QKAUJLGNZ57JD5RBXCXNCEJYQA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"92e49162515f24c272226396541aaa3bc435722bd86a26f80905e31aaef4ae5a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-16T21:37:36Z","title_canon_sha256":"3fa7ab34a340d826eec75f7a32c51b36eac1717d8fc4af217fd603530572ac85"},"schema_version":"1.0","source":{"id":"1504.04392","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.04392","created_at":"2026-05-18T01:37:13Z"},{"alias_kind":"arxiv_version","alias_value":"1504.04392v3","created_at":"2026-05-18T01:37:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04392","created_at":"2026-05-18T01:37:13Z"},{"alias_kind":"pith_short_12","alias_value":"QKAUJLGNZ57J","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QKAUJLGNZ57JD5RB","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QKAUJLGN","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:eee3cdfb4e44cde9969027342f47b0520b9962815549a9a4d4983dddfd484a5c","target":"graph","created_at":"2026-05-18T01:37:13Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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)","authors_text":"Alexander York, Talon Ward, Zi-Xia Song","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-16T21:37:36Z","title":"A Note on Weighted Rooted Trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04392","kind":"arxiv","version":3},"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:e3c0087e88c3d421dcf604e3d67253d2ec2cad643b0360cc01feb52e6a08ab0e","target":"record","created_at":"2026-05-18T01:37:13Z","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":"92e49162515f24c272226396541aaa3bc435722bd86a26f80905e31aaef4ae5a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-16T21:37:36Z","title_canon_sha256":"3fa7ab34a340d826eec75f7a32c51b36eac1717d8fc4af217fd603530572ac85"},"schema_version":"1.0","source":{"id":"1504.04392","kind":"arxiv","version":3}},"canonical_sha256":"828144accdcf7e91f621b8aed11138802f2a93fe3107a3d3d1968eb88708cb3d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"828144accdcf7e91f621b8aed11138802f2a93fe3107a3d3d1968eb88708cb3d","first_computed_at":"2026-05-18T01:37:13.334639Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:13.334639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hISGGEhwPY6mI2rMBkfw5MP3Z1/kEi4zD1gue0a4V1/oVm70pLC2cjmfd0XJ1jolW0j+Mxhv6+b10XoQeaDqAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:13.335266Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.04392","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e3c0087e88c3d421dcf604e3d67253d2ec2cad643b0360cc01feb52e6a08ab0e","sha256:eee3cdfb4e44cde9969027342f47b0520b9962815549a9a4d4983dddfd484a5c"],"state_sha256":"f110aa39717feadbc73cb8dc682d1105048bad75d33e7edc6520606c8497ce31"}