{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:I6XIFWEZQAGIPR5C2MZDHEDJI7","short_pith_number":"pith:I6XIFWEZ","canonical_record":{"source":{"id":"1101.1488","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-07T18:06:26Z","cross_cats_sorted":[],"title_canon_sha256":"e6d252457cd2c4d74d188300b7c4f6ab46cdc0adbe438404c56dc18b2e04415c","abstract_canon_sha256":"4b20596932e67fece24755b906df8f6a8f048d1760553e3f984b34b3d99ad114"},"schema_version":"1.0"},"canonical_sha256":"47ae82d899800c87c7a2d33233906947ea99c98b8ec13ede5332c0e614944dbe","source":{"kind":"arxiv","id":"1101.1488","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.1488","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"arxiv_version","alias_value":"1101.1488v1","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.1488","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"pith_short_12","alias_value":"I6XIFWEZQAGI","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"I6XIFWEZQAGIPR5C","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"I6XIFWEZ","created_at":"2026-05-18T12:26:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:I6XIFWEZQAGIPR5C2MZDHEDJI7","target":"record","payload":{"canonical_record":{"source":{"id":"1101.1488","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-07T18:06:26Z","cross_cats_sorted":[],"title_canon_sha256":"e6d252457cd2c4d74d188300b7c4f6ab46cdc0adbe438404c56dc18b2e04415c","abstract_canon_sha256":"4b20596932e67fece24755b906df8f6a8f048d1760553e3f984b34b3d99ad114"},"schema_version":"1.0"},"canonical_sha256":"47ae82d899800c87c7a2d33233906947ea99c98b8ec13ede5332c0e614944dbe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:52.773946Z","signature_b64":"43ayAh1Hl+om8yaXjNTjw1ehPE0Zh2MN3+P93JuAWpeTsUfuU9Auc6yMsMS31ZirJ7o+gd+fGlineyh9hPVHDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47ae82d899800c87c7a2d33233906947ea99c98b8ec13ede5332c0e614944dbe","last_reissued_at":"2026-05-18T04:31:52.773552Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:52.773552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1101.1488","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":"2026-05-18T04:31:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/xkv8u/NheLsd61maXJrUsO/qEzXvKO33YV1cxuQkW56bQjGrTVRNbmbaXmiicWxSj111ZsU5nyyk6o6pCPZAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:00:43.253797Z"},"content_sha256":"de4d57154c4bcef8acc6817f413f6d8366dc9de17a726b9e3402da9a0eb83e37","schema_version":"1.0","event_id":"sha256:de4d57154c4bcef8acc6817f413f6d8366dc9de17a726b9e3402da9a0eb83e37"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:I6XIFWEZQAGIPR5C2MZDHEDJI7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lines induced by bichromatic point sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Louis Theran","submitted_at":"2011-01-07T18:06:26Z","abstract_excerpt":"An important theorem of Beck says that any point set in the Euclidean plane is either ``nearly general position'' or ``nearly collinear'': there is a constant C>0 such that, given n points in the plane with at most r$ of them collinear, the number of lines induced by the points is at least Cr(n-r).\n  Recent work of Gutkin-Rams on billiards orbits requires the following elaboration of Beck's Theorem to bichromatic point sets: there is a constant C>0 such that, given n red points and n blue points in the plane with at most r of them collinear, the number of lines spanning at least one point of e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1488","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":"2026-05-18T04:31:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7/5esnism/TYfXF/5UiwL06FAQ+Ry2Bc5Ce2ydeLXQE9QcdAM1xpHtojDgCift23dEmp5CEjYFGhIPjDIk43DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:00:43.254542Z"},"content_sha256":"c26455fcecc5a691eca35f90eb6eb2cf2018c4d2e451230dfa09f674c828dfbf","schema_version":"1.0","event_id":"sha256:c26455fcecc5a691eca35f90eb6eb2cf2018c4d2e451230dfa09f674c828dfbf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I6XIFWEZQAGIPR5C2MZDHEDJI7/bundle.json","state_url":"https://pith.science/pith/I6XIFWEZQAGIPR5C2MZDHEDJI7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I6XIFWEZQAGIPR5C2MZDHEDJI7/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":"2026-06-10T23:00:43Z","links":{"resolver":"https://pith.science/pith/I6XIFWEZQAGIPR5C2MZDHEDJI7","bundle":"https://pith.science/pith/I6XIFWEZQAGIPR5C2MZDHEDJI7/bundle.json","state":"https://pith.science/pith/I6XIFWEZQAGIPR5C2MZDHEDJI7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I6XIFWEZQAGIPR5C2MZDHEDJI7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:I6XIFWEZQAGIPR5C2MZDHEDJI7","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":"4b20596932e67fece24755b906df8f6a8f048d1760553e3f984b34b3d99ad114","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-07T18:06:26Z","title_canon_sha256":"e6d252457cd2c4d74d188300b7c4f6ab46cdc0adbe438404c56dc18b2e04415c"},"schema_version":"1.0","source":{"id":"1101.1488","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.1488","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"arxiv_version","alias_value":"1101.1488v1","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.1488","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"pith_short_12","alias_value":"I6XIFWEZQAGI","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"I6XIFWEZQAGIPR5C","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"I6XIFWEZ","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:c26455fcecc5a691eca35f90eb6eb2cf2018c4d2e451230dfa09f674c828dfbf","target":"graph","created_at":"2026-05-18T04:31:52Z","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":"An important theorem of Beck says that any point set in the Euclidean plane is either ``nearly general position'' or ``nearly collinear'': there is a constant C>0 such that, given n points in the plane with at most r$ of them collinear, the number of lines induced by the points is at least Cr(n-r).\n  Recent work of Gutkin-Rams on billiards orbits requires the following elaboration of Beck's Theorem to bichromatic point sets: there is a constant C>0 such that, given n red points and n blue points in the plane with at most r of them collinear, the number of lines spanning at least one point of e","authors_text":"Louis Theran","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-07T18:06:26Z","title":"Lines induced by bichromatic point sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1488","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:de4d57154c4bcef8acc6817f413f6d8366dc9de17a726b9e3402da9a0eb83e37","target":"record","created_at":"2026-05-18T04:31:52Z","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":"4b20596932e67fece24755b906df8f6a8f048d1760553e3f984b34b3d99ad114","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-07T18:06:26Z","title_canon_sha256":"e6d252457cd2c4d74d188300b7c4f6ab46cdc0adbe438404c56dc18b2e04415c"},"schema_version":"1.0","source":{"id":"1101.1488","kind":"arxiv","version":1}},"canonical_sha256":"47ae82d899800c87c7a2d33233906947ea99c98b8ec13ede5332c0e614944dbe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47ae82d899800c87c7a2d33233906947ea99c98b8ec13ede5332c0e614944dbe","first_computed_at":"2026-05-18T04:31:52.773552Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:52.773552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"43ayAh1Hl+om8yaXjNTjw1ehPE0Zh2MN3+P93JuAWpeTsUfuU9Auc6yMsMS31ZirJ7o+gd+fGlineyh9hPVHDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:52.773946Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.1488","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de4d57154c4bcef8acc6817f413f6d8366dc9de17a726b9e3402da9a0eb83e37","sha256:c26455fcecc5a691eca35f90eb6eb2cf2018c4d2e451230dfa09f674c828dfbf"],"state_sha256":"7eda88298fc67dd6b181383165f2f5f7c9a969681b5de4dfc54797aaea1355d6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"idBTeK34EihZZbOsC12/lIL6Q+We7c9kIi+GFZQOjfHpDZ7YMVoLRUHszTO4JoEMiE8Vf+K7crfdwY4tXbH/AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T23:00:43.258416Z","bundle_sha256":"3689f0b55ec1d91b87ec6e39a7ada064bee5836f5976050d9177ea7b62e361d5"}}