{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:LHJO3FKZZAMHVAGTSARD6KVYXE","short_pith_number":"pith:LHJO3FKZ","schema_version":"1.0","canonical_sha256":"59d2ed9559c8187a80d390223f2ab8b904b38634f925ada42a59781dde770d59","source":{"kind":"arxiv","id":"1602.05798","version":1},"attestation_state":"computed","paper":{"title":"Betweenness and Nonbetweenness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ross Atkins","submitted_at":"2016-02-18T13:44:53Z","abstract_excerpt":"The betweenness function $bet(n)$ is the minimum number of total orderings of $n$ objects such that for any three distinct objects $a$, $b$ and $c$, there is an ordering in which $b$ is between $a$ and $c$. The nonbetweenness function $nbet(n)$ is the minimum number of total orderings such that for any three distinct objects $a$, $b$ and $c$, there is an ordering in which $b$ is not between $a$ and $c$. We show that $nbet(n) = \\left\\lceil \\log_2\\log_2n \\right\\rceil+1$ and $bet(n) = \\Theta(\\log n)$. Betweenness and Nonbetweenness are specific cases of a more general extreme value function calle"},"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":"1602.05798","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-18T13:44:53Z","cross_cats_sorted":[],"title_canon_sha256":"5996bd6ec10f0c2ee7f039e106358b4657a874b970d6291f9bc961efc4fc5528","abstract_canon_sha256":"0d3db032ba9e3a7cb10babc97081905e7d6b9e4b6988b54c213732c485e7a6b1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:23.198143Z","signature_b64":"D17E2PKYHahmxDQt9Q1gWcimO8lp1m93PQpm0wHaGkNCglE4UUHqaI1TASPUuxS46hCvXp5ExnHivL5cvusMAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59d2ed9559c8187a80d390223f2ab8b904b38634f925ada42a59781dde770d59","last_reissued_at":"2026-05-18T01:20:23.197503Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:23.197503Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Betweenness and Nonbetweenness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ross Atkins","submitted_at":"2016-02-18T13:44:53Z","abstract_excerpt":"The betweenness function $bet(n)$ is the minimum number of total orderings of $n$ objects such that for any three distinct objects $a$, $b$ and $c$, there is an ordering in which $b$ is between $a$ and $c$. The nonbetweenness function $nbet(n)$ is the minimum number of total orderings such that for any three distinct objects $a$, $b$ and $c$, there is an ordering in which $b$ is not between $a$ and $c$. We show that $nbet(n) = \\left\\lceil \\log_2\\log_2n \\right\\rceil+1$ and $bet(n) = \\Theta(\\log n)$. Betweenness and Nonbetweenness are specific cases of a more general extreme value function calle"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05798","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":"1602.05798","created_at":"2026-05-18T01:20:23.197657+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.05798v1","created_at":"2026-05-18T01:20:23.197657+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.05798","created_at":"2026-05-18T01:20:23.197657+00:00"},{"alias_kind":"pith_short_12","alias_value":"LHJO3FKZZAMH","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"LHJO3FKZZAMHVAGT","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"LHJO3FKZ","created_at":"2026-05-18T12:30:29.479603+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/LHJO3FKZZAMHVAGTSARD6KVYXE","json":"https://pith.science/pith/LHJO3FKZZAMHVAGTSARD6KVYXE.json","graph_json":"https://pith.science/api/pith-number/LHJO3FKZZAMHVAGTSARD6KVYXE/graph.json","events_json":"https://pith.science/api/pith-number/LHJO3FKZZAMHVAGTSARD6KVYXE/events.json","paper":"https://pith.science/paper/LHJO3FKZ"},"agent_actions":{"view_html":"https://pith.science/pith/LHJO3FKZZAMHVAGTSARD6KVYXE","download_json":"https://pith.science/pith/LHJO3FKZZAMHVAGTSARD6KVYXE.json","view_paper":"https://pith.science/paper/LHJO3FKZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.05798&json=true","fetch_graph":"https://pith.science/api/pith-number/LHJO3FKZZAMHVAGTSARD6KVYXE/graph.json","fetch_events":"https://pith.science/api/pith-number/LHJO3FKZZAMHVAGTSARD6KVYXE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LHJO3FKZZAMHVAGTSARD6KVYXE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LHJO3FKZZAMHVAGTSARD6KVYXE/action/storage_attestation","attest_author":"https://pith.science/pith/LHJO3FKZZAMHVAGTSARD6KVYXE/action/author_attestation","sign_citation":"https://pith.science/pith/LHJO3FKZZAMHVAGTSARD6KVYXE/action/citation_signature","submit_replication":"https://pith.science/pith/LHJO3FKZZAMHVAGTSARD6KVYXE/action/replication_record"}},"created_at":"2026-05-18T01:20:23.197657+00:00","updated_at":"2026-05-18T01:20:23.197657+00:00"}