{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:6NFXTX5PNN47JU7PB4RHM5LTBE","short_pith_number":"pith:6NFXTX5P","schema_version":"1.0","canonical_sha256":"f34b79dfaf6b79f4d3ef0f22767573091df6d24979b1119e1de0478ddc0ecf6a","source":{"kind":"arxiv","id":"1412.8283","version":1},"attestation_state":"computed","paper":{"title":"Lines, betweenness and metric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Cathryn Supko, Guangda Huzhang, Pierre Aboulker, Rohan Kapadia, Xiaomin Chen","submitted_at":"2014-12-29T08:52:24Z","abstract_excerpt":"A classic theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chv\\'atal conjectured that this holds for an arbitrary finite metric space, with a certain natural definition of lines in a metric space.\n  We prove that in any metric space with $n$ points, either there is a line containing all the points or there are at least $\\Omega(\\sqrt{n})$ lines. This is the first polynomial lower bound on the number of lines in general finite metric spaces. In the more general setting of pseudometric betweenness, we prove"},"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":"1412.8283","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-29T08:52:24Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"a10178241746c60a6e25b7132b24f865559fc9c52bf0982db5631b86592495de","abstract_canon_sha256":"dbd87ffc1873ca052630651df582ca1de8acf5ce6ddf5d140579d84b94feb9ed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:23.584251Z","signature_b64":"1cOaJCf0Pni7s7n8Us2OJOy9yBvfkY2JjU+Z8622F6LJ2QG1W+0tWvGA0k2lD6xKRoCTUGJlGFh96ii5hDv4Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f34b79dfaf6b79f4d3ef0f22767573091df6d24979b1119e1de0478ddc0ecf6a","last_reissued_at":"2026-05-18T02:30:23.583747Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:23.583747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lines, betweenness and metric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Cathryn Supko, Guangda Huzhang, Pierre Aboulker, Rohan Kapadia, Xiaomin Chen","submitted_at":"2014-12-29T08:52:24Z","abstract_excerpt":"A classic theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chv\\'atal conjectured that this holds for an arbitrary finite metric space, with a certain natural definition of lines in a metric space.\n  We prove that in any metric space with $n$ points, either there is a line containing all the points or there are at least $\\Omega(\\sqrt{n})$ lines. This is the first polynomial lower bound on the number of lines in general finite metric spaces. In the more general setting of pseudometric betweenness, we prove"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8283","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":"1412.8283","created_at":"2026-05-18T02:30:23.583825+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.8283v1","created_at":"2026-05-18T02:30:23.583825+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.8283","created_at":"2026-05-18T02:30:23.583825+00:00"},{"alias_kind":"pith_short_12","alias_value":"6NFXTX5PNN47","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_16","alias_value":"6NFXTX5PNN47JU7P","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_8","alias_value":"6NFXTX5P","created_at":"2026-05-18T12:28:16.859392+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/6NFXTX5PNN47JU7PB4RHM5LTBE","json":"https://pith.science/pith/6NFXTX5PNN47JU7PB4RHM5LTBE.json","graph_json":"https://pith.science/api/pith-number/6NFXTX5PNN47JU7PB4RHM5LTBE/graph.json","events_json":"https://pith.science/api/pith-number/6NFXTX5PNN47JU7PB4RHM5LTBE/events.json","paper":"https://pith.science/paper/6NFXTX5P"},"agent_actions":{"view_html":"https://pith.science/pith/6NFXTX5PNN47JU7PB4RHM5LTBE","download_json":"https://pith.science/pith/6NFXTX5PNN47JU7PB4RHM5LTBE.json","view_paper":"https://pith.science/paper/6NFXTX5P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.8283&json=true","fetch_graph":"https://pith.science/api/pith-number/6NFXTX5PNN47JU7PB4RHM5LTBE/graph.json","fetch_events":"https://pith.science/api/pith-number/6NFXTX5PNN47JU7PB4RHM5LTBE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6NFXTX5PNN47JU7PB4RHM5LTBE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6NFXTX5PNN47JU7PB4RHM5LTBE/action/storage_attestation","attest_author":"https://pith.science/pith/6NFXTX5PNN47JU7PB4RHM5LTBE/action/author_attestation","sign_citation":"https://pith.science/pith/6NFXTX5PNN47JU7PB4RHM5LTBE/action/citation_signature","submit_replication":"https://pith.science/pith/6NFXTX5PNN47JU7PB4RHM5LTBE/action/replication_record"}},"created_at":"2026-05-18T02:30:23.583825+00:00","updated_at":"2026-05-18T02:30:23.583825+00:00"}