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This settles a conjecture of Pach, Radoi\\v{c}i\\'{c}, Tardos, and T\\'oth, and yields a better bound for the famous Crossing Lemma: The crossing number, $\\mbox{cr}(G)$, of a (not too sparse) graph $G$ with $n$ vertices and $m$ edges is at least $c\\frac{m^3}{n^2}$, where $c > 1/29$. This bound is known to be tight, apart from the constant $c$ for which the previous best lower bound was $1/31.1$. As another corollary w"},"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":"1509.01932","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-07T07:24:05Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"6af2b70e5743b764cb118bf47c10370257dd81151292b438d9a092cee4d419dc","abstract_canon_sha256":"45c83cb8560a418d45a364d5b03dca47e59395aebb4e4e508cf43dee69bd5c76"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:39.025299Z","signature_b64":"LpT9M1C+5f/avMiYYN8t3QEWcNxQEtCBaTpTwWVIiiljuBFOt9Lr+8CLaWVgs6uJlGodcFZp1S2MD8h9BgzKBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0568f496de5de1e4bf13a3808e935cbabf28d75c96bf8f0d22ecc0a76e2d769c","last_reissued_at":"2026-05-17T23:50:39.024802Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:39.024802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On topological graphs with at most four crossings per edge","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Eyal Ackerman","submitted_at":"2015-09-07T07:24:05Z","abstract_excerpt":"We show that if a graph $G$ with $n \\geq 3$ vertices can be drawn in the plane such that each of its edges is involved in at most four crossings, then $G$ has at most $6n-12$ edges. This settles a conjecture of Pach, Radoi\\v{c}i\\'{c}, Tardos, and T\\'oth, and yields a better bound for the famous Crossing Lemma: The crossing number, $\\mbox{cr}(G)$, of a (not too sparse) graph $G$ with $n$ vertices and $m$ edges is at least $c\\frac{m^3}{n^2}$, where $c > 1/29$. This bound is known to be tight, apart from the constant $c$ for which the previous best lower bound was $1/31.1$. 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