{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:VDZ2SRVJKDOR7IJJHPUHPBERY6","short_pith_number":"pith:VDZ2SRVJ","canonical_record":{"source":{"id":"2605.20705","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T05:03:17Z","cross_cats_sorted":[],"title_canon_sha256":"d3502ef90b70d4911763199f5fab82928249d684db09117f40fee17a4c669918","abstract_canon_sha256":"ccf1810fd837052fb3f1c65ac7f0e99765951c21e1a005907d8c97d08a4b73c7"},"schema_version":"1.0"},"canonical_sha256":"a8f3a946a950dd1fa1293be8778491c7a3d12c41ac3996a3720a480967645058","source":{"kind":"arxiv","id":"2605.20705","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20705","created_at":"2026-05-21T01:04:50Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20705v1","created_at":"2026-05-21T01:04:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20705","created_at":"2026-05-21T01:04:50Z"},{"alias_kind":"pith_short_12","alias_value":"VDZ2SRVJKDOR","created_at":"2026-05-21T01:04:50Z"},{"alias_kind":"pith_short_16","alias_value":"VDZ2SRVJKDOR7IJJ","created_at":"2026-05-21T01:04:50Z"},{"alias_kind":"pith_short_8","alias_value":"VDZ2SRVJ","created_at":"2026-05-21T01:04:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:VDZ2SRVJKDOR7IJJHPUHPBERY6","target":"record","payload":{"canonical_record":{"source":{"id":"2605.20705","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T05:03:17Z","cross_cats_sorted":[],"title_canon_sha256":"d3502ef90b70d4911763199f5fab82928249d684db09117f40fee17a4c669918","abstract_canon_sha256":"ccf1810fd837052fb3f1c65ac7f0e99765951c21e1a005907d8c97d08a4b73c7"},"schema_version":"1.0"},"canonical_sha256":"a8f3a946a950dd1fa1293be8778491c7a3d12c41ac3996a3720a480967645058","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T01:04:50.040356Z","signature_b64":"JoMN3+AxaZQGGDWUyC6Ygkm6TBpNtAjhun92CZ1dcdCOC0Qr3df1ojQkC0S6ugwpKJbh1u/H57k3VxOc3rdbBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8f3a946a950dd1fa1293be8778491c7a3d12c41ac3996a3720a480967645058","last_reissued_at":"2026-05-21T01:04:50.039637Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T01:04:50.039637Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.20705","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":"2026-05-21T01:04:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gUftDIN2KFiKR+2+hUMs2tyDyJFKrGua699YnKOdEEyGkqd+/lWhEpiivp7nIFrDmtinXPTqVVy6CjUKrnP/AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T10:58:45.350999Z"},"content_sha256":"8f740dfa28530e95d7ff8151eb3ba3a7b26a3bc5a2ff199ae855e7153fb34afd","schema_version":"1.0","event_id":"sha256:8f740dfa28530e95d7ff8151eb3ba3a7b26a3bc5a2ff199ae855e7153fb34afd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:VDZ2SRVJKDOR7IJJHPUHPBERY6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Extremal structure in dense arrangements of $k$-intersecting curves","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Suk, Su Zhou","submitted_at":"2026-05-20T05:03:17Z","abstract_excerpt":"Let $P$ be a set of $n$ points in the plane, and let $\\mathcal C$ be a collection of $n$ simple $k$-intersecting curves, meaning that every two distinct curves of $\\mathcal C$ meet in at most $k$ points. A classical theorem of Pach and Sharir from 1998 gives the upper bound $I(P,\\mathcal C)=O_k(n^{(3k+1)/(2k+1)})$. We prove that this bound can be improved when one excludes a complete local incidence pattern. More precisely, for any fixed integers $s>k+1\\ge 2$, if there do not exist $s$ points of $P$ such that every $(k+1)$-tuple among them is contained in a distinct curve of $\\mathcal C$, then"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20705","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.20705/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-05-21T01:04:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"prHovpU2fGcj17bDm5MXV8q1W6szCZq3FdcHhGXOg9oR/iqSeJEbAdftDYuEwih7Bcq0tcGuJgiBaYTGPUBCCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T10:58:45.351635Z"},"content_sha256":"dc9b3ce61cd8d05ef55013fd113d0b549850796ca65c32fe856c1e4b599c7998","schema_version":"1.0","event_id":"sha256:dc9b3ce61cd8d05ef55013fd113d0b549850796ca65c32fe856c1e4b599c7998"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VDZ2SRVJKDOR7IJJHPUHPBERY6/bundle.json","state_url":"https://pith.science/pith/VDZ2SRVJKDOR7IJJHPUHPBERY6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VDZ2SRVJKDOR7IJJHPUHPBERY6/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":"2026-05-24T10:58:45Z","links":{"resolver":"https://pith.science/pith/VDZ2SRVJKDOR7IJJHPUHPBERY6","bundle":"https://pith.science/pith/VDZ2SRVJKDOR7IJJHPUHPBERY6/bundle.json","state":"https://pith.science/pith/VDZ2SRVJKDOR7IJJHPUHPBERY6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VDZ2SRVJKDOR7IJJHPUHPBERY6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VDZ2SRVJKDOR7IJJHPUHPBERY6","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":"ccf1810fd837052fb3f1c65ac7f0e99765951c21e1a005907d8c97d08a4b73c7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T05:03:17Z","title_canon_sha256":"d3502ef90b70d4911763199f5fab82928249d684db09117f40fee17a4c669918"},"schema_version":"1.0","source":{"id":"2605.20705","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20705","created_at":"2026-05-21T01:04:50Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20705v1","created_at":"2026-05-21T01:04:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20705","created_at":"2026-05-21T01:04:50Z"},{"alias_kind":"pith_short_12","alias_value":"VDZ2SRVJKDOR","created_at":"2026-05-21T01:04:50Z"},{"alias_kind":"pith_short_16","alias_value":"VDZ2SRVJKDOR7IJJ","created_at":"2026-05-21T01:04:50Z"},{"alias_kind":"pith_short_8","alias_value":"VDZ2SRVJ","created_at":"2026-05-21T01:04:50Z"}],"graph_snapshots":[{"event_id":"sha256:dc9b3ce61cd8d05ef55013fd113d0b549850796ca65c32fe856c1e4b599c7998","target":"graph","created_at":"2026-05-21T01:04:50Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.20705/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $P$ be a set of $n$ points in the plane, and let $\\mathcal C$ be a collection of $n$ simple $k$-intersecting curves, meaning that every two distinct curves of $\\mathcal C$ meet in at most $k$ points. A classical theorem of Pach and Sharir from 1998 gives the upper bound $I(P,\\mathcal C)=O_k(n^{(3k+1)/(2k+1)})$. We prove that this bound can be improved when one excludes a complete local incidence pattern. More precisely, for any fixed integers $s>k+1\\ge 2$, if there do not exist $s$ points of $P$ such that every $(k+1)$-tuple among them is contained in a distinct curve of $\\mathcal C$, then","authors_text":"Andrew Suk, Su Zhou","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T05:03:17Z","title":"Extremal structure in dense arrangements of $k$-intersecting curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20705","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":"sha256:8f740dfa28530e95d7ff8151eb3ba3a7b26a3bc5a2ff199ae855e7153fb34afd","target":"record","created_at":"2026-05-21T01:04:50Z","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":"ccf1810fd837052fb3f1c65ac7f0e99765951c21e1a005907d8c97d08a4b73c7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T05:03:17Z","title_canon_sha256":"d3502ef90b70d4911763199f5fab82928249d684db09117f40fee17a4c669918"},"schema_version":"1.0","source":{"id":"2605.20705","kind":"arxiv","version":1}},"canonical_sha256":"a8f3a946a950dd1fa1293be8778491c7a3d12c41ac3996a3720a480967645058","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a8f3a946a950dd1fa1293be8778491c7a3d12c41ac3996a3720a480967645058","first_computed_at":"2026-05-21T01:04:50.039637Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-21T01:04:50.039637Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JoMN3+AxaZQGGDWUyC6Ygkm6TBpNtAjhun92CZ1dcdCOC0Qr3df1ojQkC0S6ugwpKJbh1u/H57k3VxOc3rdbBg==","signature_status":"signed_v1","signed_at":"2026-05-21T01:04:50.040356Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.20705","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8f740dfa28530e95d7ff8151eb3ba3a7b26a3bc5a2ff199ae855e7153fb34afd","sha256:dc9b3ce61cd8d05ef55013fd113d0b549850796ca65c32fe856c1e4b599c7998"],"state_sha256":"1b9cf581d36e78d0de05c172e9cd33710d5c2adc436212b3097f02b02ddd8f53"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"570OAiCVBK2HqU0f2RlywzVpaj8O2iN/RSnUm+6e0kCw8cFsuZUgZwEG5QKLaLjmo+vatzNB1uPzQJGjKaHSDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-24T10:58:45.355258Z","bundle_sha256":"ccfae38516d005ef090858c8e07cc376f3aa322a4d1b28058f402505ce8f8f07"}}