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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"},"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":"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"},"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"},"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. 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