{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:ZZCFCBXQJGO5KJOHR2NT764KRL","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":"8e1cdc3cb8c665a001b76b220e00ea5e0ea600b19fce1d121b4c83b97343d26a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2010-07-15T09:59:58Z","title_canon_sha256":"c6a40cb53c98ac748447a4d61ec2daa220f89e17e263a1a6167de10264ac3a42"},"schema_version":"1.0","source":{"id":"1007.2518","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.2518","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"arxiv_version","alias_value":"1007.2518v2","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2518","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"pith_short_12","alias_value":"ZZCFCBXQJGO5","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"ZZCFCBXQJGO5KJOH","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"ZZCFCBXQ","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:c46c0642c2d0f1f5788d4a5c0f4506f7880b6b23f1f6d7cf6e3a7b63c138f5ad","target":"graph","created_at":"2026-05-18T04:42:22Z","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":"Borsuk conjectured that every n-dimensional bounded set of positive diameter can be partitioned into n+1 sets of smaller diameters. This conjecture was proved for n=2 by Borsuk, for n=3 first by Eggleston, and disproved for n > 297 by Hinrichs and Richer. It is not known if the conjecture holds for 3 < n < 298. The best upper bound for the number of subsets of smaller diameters a four-dimensional set can be partitioned into is nine. This estimate was given by Lassak in 1982. In this note we improve this estimate by one.","authors_text":"Zsolt Langi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2010-07-15T09:59:58Z","title":"On the Borsuk number of four-dimensional sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2518","kind":"arxiv","version":2},"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:dc8c89720748c786a06606bd4dc038670fc256f9973fb61e309db1f036a9402e","target":"record","created_at":"2026-05-18T04:42:22Z","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":"8e1cdc3cb8c665a001b76b220e00ea5e0ea600b19fce1d121b4c83b97343d26a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2010-07-15T09:59:58Z","title_canon_sha256":"c6a40cb53c98ac748447a4d61ec2daa220f89e17e263a1a6167de10264ac3a42"},"schema_version":"1.0","source":{"id":"1007.2518","kind":"arxiv","version":2}},"canonical_sha256":"ce445106f0499dd525c78e9b3ffb8a8ad9cb04c8069c0ebc79d9f704ebc28f80","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce445106f0499dd525c78e9b3ffb8a8ad9cb04c8069c0ebc79d9f704ebc28f80","first_computed_at":"2026-05-18T04:42:22.986628Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:42:22.986628Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"D/+Doogrf46rJo8uoZk8xwtxucnoTBrcZ+D6TrZxbBi55fNiFcoxX2uD1hQrPX/25+v+CAaB496iJQaeW+R+AA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:42:22.987164Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.2518","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc8c89720748c786a06606bd4dc038670fc256f9973fb61e309db1f036a9402e","sha256:c46c0642c2d0f1f5788d4a5c0f4506f7880b6b23f1f6d7cf6e3a7b63c138f5ad"],"state_sha256":"956dc2153231c27c143a86eb77d0e7b674be34ea50469a4b0e758bc98ecbfe31"}