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The main result of this paper is a Crossing Lemma for simple curves: Let $X$ and $T$ stand for the sets of intersection points and touching points, respectively, in a family of $n$ simple curves in the plane, no three of which pass through the same point. If $|T|>cn$, for some fixed constant $c>0$, then we prove that $|X|=\\Omega(|T|(\\log\\log(|T|/n))^{1/504})$. In particular, if $|T|/n\\rightarrow\\infty$, then the number of intersecti"},"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":"1708.02077","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-07T11:40:29Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"0516cd7d725b270ba487318b2bb787a791255ee8df7339cfb52475895ba3ea59","abstract_canon_sha256":"afc2b5005ae7a414d78e28950e6d98eb73fd31a861a86996f0e924ca495fde47"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:32.369194Z","signature_b64":"1gl11wnm47KWJouBaNapr42uUO3nP2jZj27eFAaPbD+yOv/3/1kNFaNg4iZqQnqeOBbortIjAF2STM1lPxrDDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7cc508b147e93bebb9ab2ee6535fb099ea2ffa1f61d62e7ca00cdbeb26cba287","last_reissued_at":"2026-05-18T00:38:32.368742Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:32.368742Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Crossing Lemma for Jordan Curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"G\\'abor Tardos, J\\'anos Pach, Natan Rubin","submitted_at":"2017-08-07T11:40:29Z","abstract_excerpt":"If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a {\\em touching point}. The main result of this paper is a Crossing Lemma for simple curves: Let $X$ and $T$ stand for the sets of intersection points and touching points, respectively, in a family of $n$ simple curves in the plane, no three of which pass through the same point. If $|T|>cn$, for some fixed constant $c>0$, then we prove that $|X|=\\Omega(|T|(\\log\\log(|T|/n))^{1/504})$. 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