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[\\emph{Computational Geometry: Theory and Applications} {\\bf 68} 2--6, (2018)] showed that for every $k\\ge 1$, we have ${\\mbox {cr}}_k(G)\\le \\left(\\frac{2}{k^2}-\\frac1{k^3}\\right){\\mbox {cr}}(G)$ and that thi"},"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":"1807.03430","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-10T00:41:13Z","cross_cats_sorted":[],"title_canon_sha256":"1d283cb5cfdc3ddc58550dc130cdd633cc34c968101b111284ae75e5b7de9862","abstract_canon_sha256":"ae9f75ab2ccdf59822101b42cf771d3ec119c7fc33b9274ede5887c81bc5ffe5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:10.459678Z","signature_b64":"SqqtmQA1lGqX6f+p8y3FP8+BukTaFqvnmoJhfVjOCnUI9zYV/Nc9Rx10QjOOPbqpb0ttMeWbC2tZP9kDECuaCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2b59b652533d567bb2528007ff66db11d400ab54466cee38f6aa00c243c8edf","last_reissued_at":"2026-05-18T00:11:10.459018Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:10.459018Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Using Block Designs in Crossing Number Bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Arran Hamm, Eva Czabarka, Garner Cochran, Gregory Clark, Gwen Spencer, John Asplund, Laszlo Szekely, Libby Taylor, Zhiyu Wang","submitted_at":"2018-07-10T00:41:13Z","abstract_excerpt":"The crossing number ${\\mbox {cr}}(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. 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