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The complex K is d-representable if there is a collection {C_1,...,C_n} of convex sets in R^d such that a subcollection {C_{i_1},...,C_{i_j}} has a nonempty intersection if and only if {v_{i_1},...,v_{i_j}} is a face of K.\n  In 1967 Wegner proved that every simplicial complex of dimension d is (2d+1)-representable. He also suggested that his bound is the best possible, i.e., that there are $d$-dimensional simplicial complexes which are not 2d-representable. However, he was not able to prove his suggestion.\n  We prove that his su"},"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":"1107.1170","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-06T15:59:02Z","cross_cats_sorted":[],"title_canon_sha256":"82218b9f47f552433fce234f943df27cda9ef8cbd279a19f574ba1e943e32a2a","abstract_canon_sha256":"0025c05fb67902234e4e8f271c15b9e9b055b072fd643cab81d51c0e0c76e825"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:45.431029Z","signature_b64":"PlDypOSPcfA8JAPGKNs/UDtj8pLg9zzEH737ddDvjik2CldLxEO2jxN5SofkrMpMifSKNFyPEMf943eOsT19Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1940642133e2584c0ba9e00125abf5d2ebff90d5ad97dba23d8d5b5f5af13199","last_reissued_at":"2026-05-18T04:18:45.430409Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:45.430409Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"d-Representability of simplicial complexes of fixed dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Martin Tancer","submitted_at":"2011-07-06T15:59:02Z","abstract_excerpt":"Let K be a simplicial complex with vertex set V = {v_1,..., v_n}. The complex K is d-representable if there is a collection {C_1,...,C_n} of convex sets in R^d such that a subcollection {C_{i_1},...,C_{i_j}} has a nonempty intersection if and only if {v_{i_1},...,v_{i_j}} is a face of K.\n  In 1967 Wegner proved that every simplicial complex of dimension d is (2d+1)-representable. He also suggested that his bound is the best possible, i.e., that there are $d$-dimensional simplicial complexes which are not 2d-representable. However, he was not able to prove his suggestion.\n  We prove that his su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.1170","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1107.1170","created_at":"2026-05-18T04:18:45.430515+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.1170v1","created_at":"2026-05-18T04:18:45.430515+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.1170","created_at":"2026-05-18T04:18:45.430515+00:00"},{"alias_kind":"pith_short_12","alias_value":"DFAGIIJT4JME","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"DFAGIIJT4JMEYC5J","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"DFAGIIJT","created_at":"2026-05-18T12:26:26.731475+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DFAGIIJT4JMEYC5J4AASLK7V2L","json":"https://pith.science/pith/DFAGIIJT4JMEYC5J4AASLK7V2L.json","graph_json":"https://pith.science/api/pith-number/DFAGIIJT4JMEYC5J4AASLK7V2L/graph.json","events_json":"https://pith.science/api/pith-number/DFAGIIJT4JMEYC5J4AASLK7V2L/events.json","paper":"https://pith.science/paper/DFAGIIJT"},"agent_actions":{"view_html":"https://pith.science/pith/DFAGIIJT4JMEYC5J4AASLK7V2L","download_json":"https://pith.science/pith/DFAGIIJT4JMEYC5J4AASLK7V2L.json","view_paper":"https://pith.science/paper/DFAGIIJT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.1170&json=true","fetch_graph":"https://pith.science/api/pith-number/DFAGIIJT4JMEYC5J4AASLK7V2L/graph.json","fetch_events":"https://pith.science/api/pith-number/DFAGIIJT4JMEYC5J4AASLK7V2L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DFAGIIJT4JMEYC5J4AASLK7V2L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DFAGIIJT4JMEYC5J4AASLK7V2L/action/storage_attestation","attest_author":"https://pith.science/pith/DFAGIIJT4JMEYC5J4AASLK7V2L/action/author_attestation","sign_citation":"https://pith.science/pith/DFAGIIJT4JMEYC5J4AASLK7V2L/action/citation_signature","submit_replication":"https://pith.science/pith/DFAGIIJT4JMEYC5J4AASLK7V2L/action/replication_record"}},"created_at":"2026-05-18T04:18:45.430515+00:00","updated_at":"2026-05-18T04:18:45.430515+00:00"}