{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:FKTEJWBHONB3TJJYPJ4ERTPWNB","short_pith_number":"pith:FKTEJWBH","canonical_record":{"source":{"id":"1412.1060","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-02T20:45:22Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"ee6f8ff8b201bc5a9f3114cb656d72b92729fb897a92308b59a7b3a2ececd45d","abstract_canon_sha256":"9016998829612d7c67c5d1bb4b6507eef7b3fa4dce496c6ba36c1b8ea976132a"},"schema_version":"1.0"},"canonical_sha256":"2aa644d8277343b9a5387a7848cdf6687d22e4a1dc356afb123be25619013fc6","source":{"kind":"arxiv","id":"1412.1060","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.1060","created_at":"2026-05-18T02:32:15Z"},{"alias_kind":"arxiv_version","alias_value":"1412.1060v1","created_at":"2026-05-18T02:32:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.1060","created_at":"2026-05-18T02:32:15Z"},{"alias_kind":"pith_short_12","alias_value":"FKTEJWBHONB3","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"FKTEJWBHONB3TJJY","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"FKTEJWBH","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:FKTEJWBHONB3TJJYPJ4ERTPWNB","target":"record","payload":{"canonical_record":{"source":{"id":"1412.1060","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-02T20:45:22Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"ee6f8ff8b201bc5a9f3114cb656d72b92729fb897a92308b59a7b3a2ececd45d","abstract_canon_sha256":"9016998829612d7c67c5d1bb4b6507eef7b3fa4dce496c6ba36c1b8ea976132a"},"schema_version":"1.0"},"canonical_sha256":"2aa644d8277343b9a5387a7848cdf6687d22e4a1dc356afb123be25619013fc6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:15.014145Z","signature_b64":"Ki87s4DOOkRjYlb/T8dDA7zMZ0wucNY3oCpTrRxy/7szYi4DlQ4Y/CWOuQ3vG/DB5nDwT6WJTz5MItNLEPYwBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2aa644d8277343b9a5387a7848cdf6687d22e4a1dc356afb123be25619013fc6","last_reissued_at":"2026-05-18T02:32:15.013387Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:15.013387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.1060","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:32:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wj2EmVgItbfKYwd0tvrSut63g0by0gUQt2HTZCSndt3ksTx1Hf9gpTfLovc7F7GEwZlw4tTOy8hivz+36bhEBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T00:10:44.161096Z"},"content_sha256":"721b6a2457c96a8e39f8297413fe8dc91ac965a00851d0826e20712e678d5ac0","schema_version":"1.0","event_id":"sha256:721b6a2457c96a8e39f8297413fe8dc91ac965a00851d0826e20712e678d5ac0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:FKTEJWBHONB3TJJYPJ4ERTPWNB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the number of rich lines in truly high dimensional sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Sivakanth Gopi, Zeev Dvir","submitted_at":"2014-12-02T20:45:22Z","abstract_excerpt":"We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a `truly' $d$-dimensional configuration of points $v_1,\\ldots,v_n \\in \\mathbb{C}^d$. More formally, we show that, if the number of $r$-rich lines is significantly larger than $n^2/r^d$ then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor $r^d$ can be replaced with a tight $r^{d+1}$. If true, this would generalize the classic Szemer\\'edi-Trotter theorem which gives a bound of $n^2/r^3$ on the number of $r$-rich lines in a planar configuration. Thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.1060","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:32:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HDMtwkSqimwQkcI/ZdMbYSemqVUv2z9zEgWnA6lWxNvAlCAklduIz3IDvMn3mTnweBsDvcLZ+E3C45NftyD2AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T00:10:44.161862Z"},"content_sha256":"86f96ff912c2e974efaf06168940cfee6e1fd547aa3104e5be1be1ab7a3c3081","schema_version":"1.0","event_id":"sha256:86f96ff912c2e974efaf06168940cfee6e1fd547aa3104e5be1be1ab7a3c3081"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FKTEJWBHONB3TJJYPJ4ERTPWNB/bundle.json","state_url":"https://pith.science/pith/FKTEJWBHONB3TJJYPJ4ERTPWNB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FKTEJWBHONB3TJJYPJ4ERTPWNB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-30T00:10:44Z","links":{"resolver":"https://pith.science/pith/FKTEJWBHONB3TJJYPJ4ERTPWNB","bundle":"https://pith.science/pith/FKTEJWBHONB3TJJYPJ4ERTPWNB/bundle.json","state":"https://pith.science/pith/FKTEJWBHONB3TJJYPJ4ERTPWNB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FKTEJWBHONB3TJJYPJ4ERTPWNB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:FKTEJWBHONB3TJJYPJ4ERTPWNB","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":"9016998829612d7c67c5d1bb4b6507eef7b3fa4dce496c6ba36c1b8ea976132a","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-02T20:45:22Z","title_canon_sha256":"ee6f8ff8b201bc5a9f3114cb656d72b92729fb897a92308b59a7b3a2ececd45d"},"schema_version":"1.0","source":{"id":"1412.1060","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.1060","created_at":"2026-05-18T02:32:15Z"},{"alias_kind":"arxiv_version","alias_value":"1412.1060v1","created_at":"2026-05-18T02:32:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.1060","created_at":"2026-05-18T02:32:15Z"},{"alias_kind":"pith_short_12","alias_value":"FKTEJWBHONB3","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"FKTEJWBHONB3TJJY","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"FKTEJWBH","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:86f96ff912c2e974efaf06168940cfee6e1fd547aa3104e5be1be1ab7a3c3081","target":"graph","created_at":"2026-05-18T02:32:15Z","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":"We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a `truly' $d$-dimensional configuration of points $v_1,\\ldots,v_n \\in \\mathbb{C}^d$. More formally, we show that, if the number of $r$-rich lines is significantly larger than $n^2/r^d$ then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor $r^d$ can be replaced with a tight $r^{d+1}$. If true, this would generalize the classic Szemer\\'edi-Trotter theorem which gives a bound of $n^2/r^3$ on the number of $r$-rich lines in a planar configuration. Thi","authors_text":"Sivakanth Gopi, Zeev Dvir","cross_cats":["cs.CG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-02T20:45:22Z","title":"On the number of rich lines in truly high dimensional sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.1060","kind":"arxiv","version":1},"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":"sha256:721b6a2457c96a8e39f8297413fe8dc91ac965a00851d0826e20712e678d5ac0","target":"record","created_at":"2026-05-18T02:32:15Z","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":"9016998829612d7c67c5d1bb4b6507eef7b3fa4dce496c6ba36c1b8ea976132a","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-02T20:45:22Z","title_canon_sha256":"ee6f8ff8b201bc5a9f3114cb656d72b92729fb897a92308b59a7b3a2ececd45d"},"schema_version":"1.0","source":{"id":"1412.1060","kind":"arxiv","version":1}},"canonical_sha256":"2aa644d8277343b9a5387a7848cdf6687d22e4a1dc356afb123be25619013fc6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2aa644d8277343b9a5387a7848cdf6687d22e4a1dc356afb123be25619013fc6","first_computed_at":"2026-05-18T02:32:15.013387Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:15.013387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ki87s4DOOkRjYlb/T8dDA7zMZ0wucNY3oCpTrRxy/7szYi4DlQ4Y/CWOuQ3vG/DB5nDwT6WJTz5MItNLEPYwBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:15.014145Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.1060","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:721b6a2457c96a8e39f8297413fe8dc91ac965a00851d0826e20712e678d5ac0","sha256:86f96ff912c2e974efaf06168940cfee6e1fd547aa3104e5be1be1ab7a3c3081"],"state_sha256":"bf3f0a3fb549b084c46e9e13e18c820c914b5485d02ffdc57484f9989ea8db5a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hSfSm31Dw7OT8vKvZtI6NAnBCediW1qh4BDyfkClgqhJrDaEQWnwxL3H/e4+FCjDpGklaOyMw/CKM2YmGitdAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T00:10:44.165664Z","bundle_sha256":"14307cd72cf837fd32e778695a6c5f7be3b5b6d8bb9515ebc70553de461500f8"}}