{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:YK2ZWZJFGPKWPOZFFAAH75TNWE","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":"ae9f75ab2ccdf59822101b42cf771d3ec119c7fc33b9274ede5887c81bc5ffe5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-10T00:41:13Z","title_canon_sha256":"1d283cb5cfdc3ddc58550dc130cdd633cc34c968101b111284ae75e5b7de9862"},"schema_version":"1.0","source":{"id":"1807.03430","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.03430","created_at":"2026-05-18T00:11:10Z"},{"alias_kind":"arxiv_version","alias_value":"1807.03430v1","created_at":"2026-05-18T00:11:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.03430","created_at":"2026-05-18T00:11:10Z"},{"alias_kind":"pith_short_12","alias_value":"YK2ZWZJFGPKW","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"YK2ZWZJFGPKWPOZF","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"YK2ZWZJF","created_at":"2026-05-18T12:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:626db3b4bf061b7184193792696be35b52776fd2930d7b6a6ac3d169c2a92761","target":"graph","created_at":"2026-05-18T00:11:10Z","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"},"paper":{"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. For any $k\\ge 1$, the $k$-planar crossing number of $G$, ${\\mbox {cr}}_k(G)$, is defined as the minimum of ${\\mbox {cr}}(G_1)+{\\mbox {cr}}(G_2)+\\ldots+{\\mbox {cr}}(G_{k})$ over all graphs $G_1, G_2,\\ldots, G_{k}$ with $\\cup_{i=1}^{k}G_i=G$. Pach et al. [\\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","authors_text":"Arran Hamm, Eva Czabarka, Garner Cochran, Gregory Clark, Gwen Spencer, John Asplund, Laszlo Szekely, Libby Taylor, Zhiyu Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-10T00:41:13Z","title":"Using Block Designs in Crossing Number Bounds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.03430","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:dc5064fdd468ba62e23f40c7915ecbece7b2d3d1e13cbcd636dc53c97ce75a69","target":"record","created_at":"2026-05-18T00:11:10Z","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":"ae9f75ab2ccdf59822101b42cf771d3ec119c7fc33b9274ede5887c81bc5ffe5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-10T00:41:13Z","title_canon_sha256":"1d283cb5cfdc3ddc58550dc130cdd633cc34c968101b111284ae75e5b7de9862"},"schema_version":"1.0","source":{"id":"1807.03430","kind":"arxiv","version":1}},"canonical_sha256":"c2b59b652533d567bb2528007ff66db11d400ab54466cee38f6aa00c243c8edf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c2b59b652533d567bb2528007ff66db11d400ab54466cee38f6aa00c243c8edf","first_computed_at":"2026-05-18T00:11:10.459018Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:11:10.459018Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SqqtmQA1lGqX6f+p8y3FP8+BukTaFqvnmoJhfVjOCnUI9zYV/Nc9Rx10QjOOPbqpb0ttMeWbC2tZP9kDECuaCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:11:10.459678Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.03430","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc5064fdd468ba62e23f40c7915ecbece7b2d3d1e13cbcd636dc53c97ce75a69","sha256:626db3b4bf061b7184193792696be35b52776fd2930d7b6a6ac3d169c2a92761"],"state_sha256":"d65fee5c66454c103a8a372b12106f7f64c7f76c7e4183f108562e71a530dcd5"}