{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:7GA7VYM6K2AD6N6RIOVMB4ATM5","short_pith_number":"pith:7GA7VYM6","canonical_record":{"source":{"id":"1805.09888","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-24T20:33:20Z","cross_cats_sorted":[],"title_canon_sha256":"5e734c2b30ae3ef90a7565663687d42599fe15c570510228805a9d87cf4a8a09","abstract_canon_sha256":"e9574ee3501ae7d8117aa054a6601a0302b00692c4251d0924e04fff21aa941b"},"schema_version":"1.0"},"canonical_sha256":"f981fae19e56803f37d143aac0f013677b837ce5b403030f0e9ad5f744cfe4c6","source":{"kind":"arxiv","id":"1805.09888","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.09888","created_at":"2026-05-17T23:49:22Z"},{"alias_kind":"arxiv_version","alias_value":"1805.09888v2","created_at":"2026-05-17T23:49:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.09888","created_at":"2026-05-17T23:49:22Z"},{"alias_kind":"pith_short_12","alias_value":"7GA7VYM6K2AD","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"7GA7VYM6K2AD6N6R","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"7GA7VYM6","created_at":"2026-05-18T12:32:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:7GA7VYM6K2AD6N6RIOVMB4ATM5","target":"record","payload":{"canonical_record":{"source":{"id":"1805.09888","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-24T20:33:20Z","cross_cats_sorted":[],"title_canon_sha256":"5e734c2b30ae3ef90a7565663687d42599fe15c570510228805a9d87cf4a8a09","abstract_canon_sha256":"e9574ee3501ae7d8117aa054a6601a0302b00692c4251d0924e04fff21aa941b"},"schema_version":"1.0"},"canonical_sha256":"f981fae19e56803f37d143aac0f013677b837ce5b403030f0e9ad5f744cfe4c6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:22.462753Z","signature_b64":"1gt4Mr2ClJ109Pv7hfbSGbuzEzJx+F4iTFolwNwGTFlxV1RH+Pw8zarzSFQGsASVeQeVZCNjA8Tlgw4y20+0CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f981fae19e56803f37d143aac0f013677b837ce5b403030f0e9ad5f744cfe4c6","last_reissued_at":"2026-05-17T23:49:22.462101Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:22.462101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.09888","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:49:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PgtzCPdDyMLVV5e6/ZoJZ6Z9MU4OaLTXFxhhNQG1SYJqAd3Dhsva/TEOY7kI08WGjDqyx3TsIq/wOGioNb8OAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T17:10:22.790695Z"},"content_sha256":"716d76655e8c7a190cf15791618f05aff7daa5e384d85ad2e1fdf36bc1465a35","schema_version":"1.0","event_id":"sha256:716d76655e8c7a190cf15791618f05aff7daa5e384d85ad2e1fdf36bc1465a35"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:7GA7VYM6K2AD6N6RIOVMB4ATM5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On crossing families of complete geometric graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Rubio-Montiel, Dolores Lara","submitted_at":"2018-05-24T20:33:20Z","abstract_excerpt":"A crossing family is a collection of pairwise crossing segments, this concept was introduced by Aronov et. al. (1994). They prove that any set of $n$ points (in general position) in the plain contains a crossing family of size $\\sqrt{n/12}$. In this paper we present a generalization of the concept and give several results regarding this generalization."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09888","kind":"arxiv","version":2},"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":"2026-05-17T23:49:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jQwAQgCBZP+91YGI5++yIZhjEeOtOCPPdwu4377jdX0bg1sNYCtTL4cKPaqepyLeQzcg2d8e6tfVzP4BQYMNBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T17:10:22.791036Z"},"content_sha256":"d90420378207d3ca748d053c1edaf40d3c76550c0914bbe0a365beb34fffd2fa","schema_version":"1.0","event_id":"sha256:d90420378207d3ca748d053c1edaf40d3c76550c0914bbe0a365beb34fffd2fa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7GA7VYM6K2AD6N6RIOVMB4ATM5/bundle.json","state_url":"https://pith.science/pith/7GA7VYM6K2AD6N6RIOVMB4ATM5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7GA7VYM6K2AD6N6RIOVMB4ATM5/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":"2026-06-25T17:10:22Z","links":{"resolver":"https://pith.science/pith/7GA7VYM6K2AD6N6RIOVMB4ATM5","bundle":"https://pith.science/pith/7GA7VYM6K2AD6N6RIOVMB4ATM5/bundle.json","state":"https://pith.science/pith/7GA7VYM6K2AD6N6RIOVMB4ATM5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7GA7VYM6K2AD6N6RIOVMB4ATM5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:7GA7VYM6K2AD6N6RIOVMB4ATM5","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":"e9574ee3501ae7d8117aa054a6601a0302b00692c4251d0924e04fff21aa941b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-24T20:33:20Z","title_canon_sha256":"5e734c2b30ae3ef90a7565663687d42599fe15c570510228805a9d87cf4a8a09"},"schema_version":"1.0","source":{"id":"1805.09888","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.09888","created_at":"2026-05-17T23:49:22Z"},{"alias_kind":"arxiv_version","alias_value":"1805.09888v2","created_at":"2026-05-17T23:49:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.09888","created_at":"2026-05-17T23:49:22Z"},{"alias_kind":"pith_short_12","alias_value":"7GA7VYM6K2AD","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"7GA7VYM6K2AD6N6R","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"7GA7VYM6","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:d90420378207d3ca748d053c1edaf40d3c76550c0914bbe0a365beb34fffd2fa","target":"graph","created_at":"2026-05-17T23:49:22Z","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":"A crossing family is a collection of pairwise crossing segments, this concept was introduced by Aronov et. al. (1994). They prove that any set of $n$ points (in general position) in the plain contains a crossing family of size $\\sqrt{n/12}$. In this paper we present a generalization of the concept and give several results regarding this generalization.","authors_text":"Christian Rubio-Montiel, Dolores Lara","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-24T20:33:20Z","title":"On crossing families of complete geometric graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09888","kind":"arxiv","version":2},"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":"sha256:716d76655e8c7a190cf15791618f05aff7daa5e384d85ad2e1fdf36bc1465a35","target":"record","created_at":"2026-05-17T23:49:22Z","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":"e9574ee3501ae7d8117aa054a6601a0302b00692c4251d0924e04fff21aa941b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-24T20:33:20Z","title_canon_sha256":"5e734c2b30ae3ef90a7565663687d42599fe15c570510228805a9d87cf4a8a09"},"schema_version":"1.0","source":{"id":"1805.09888","kind":"arxiv","version":2}},"canonical_sha256":"f981fae19e56803f37d143aac0f013677b837ce5b403030f0e9ad5f744cfe4c6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f981fae19e56803f37d143aac0f013677b837ce5b403030f0e9ad5f744cfe4c6","first_computed_at":"2026-05-17T23:49:22.462101Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:49:22.462101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1gt4Mr2ClJ109Pv7hfbSGbuzEzJx+F4iTFolwNwGTFlxV1RH+Pw8zarzSFQGsASVeQeVZCNjA8Tlgw4y20+0CA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:49:22.462753Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.09888","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:716d76655e8c7a190cf15791618f05aff7daa5e384d85ad2e1fdf36bc1465a35","sha256:d90420378207d3ca748d053c1edaf40d3c76550c0914bbe0a365beb34fffd2fa"],"state_sha256":"466f4cf7d70c51589b63f3e2163f2420d718343ac3594982655f886427f582ac"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PELWuL/wa5/n+PcIAu6wgjxLCPeHP6DlUX9lPhKLxscKJF6xxuAaUUv8hISaFP7wOYRs/vrUg9n67r13BiDSCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T17:10:22.792949Z","bundle_sha256":"dabf5c97067ec5d9fadf460e996c68ab0bd05a12811a56b09e224a4e632ee66b"}}