{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:ADBGHYG3VFDVPSHBBTLUCWUETL","short_pith_number":"pith:ADBGHYG3","schema_version":"1.0","canonical_sha256":"00c263e0dba94757c8e10cd7415a849ac203ac510fa422ed0d00f8089f1dd103","source":{"kind":"arxiv","id":"1908.07097","version":2},"attestation_state":"computed","paper":{"title":"An Omega(n^2) Lower Bound for Random Universal Sets for Planar Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","math.CO"],"primary_cat":"cs.DM","authors_text":"Alexander Choi, Kevin Costello, Marek Chrobak","submitted_at":"2019-08-19T22:57:53Z","abstract_excerpt":"A set $U\\subseteq \\reals^2$ is $n$-universal if all $n$-vertex planar graphs have a planar straight-line embedding into $U$. We prove that if $Q \\subseteq \\reals^2$ consists of points chosen randomly and uniformly from the unit square then $Q$ must have cardinality $\\Omega(n^2)$ in order to be $n$-universal with high probability. This shows that the probabilistic method, at least in its basic form, cannot be used to establish an $o(n^2)$ upper bound on universal sets."},"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":"1908.07097","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2019-08-19T22:57:53Z","cross_cats_sorted":["cs.CG","math.CO"],"title_canon_sha256":"ef3feb39a273c70a165442b2feaf56816f977b565e142b2de21233bebe575a56","abstract_canon_sha256":"167526aba8878f608e2e8d65144b6424a7446719538e49f8afab7b54f92d9578"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:03:58.273667Z","signature_b64":"DZ7xAgly1yR/ttdjCqg1npY9KQ6aW/qX5JGfK7/lvaqvCdHLuKpBM26T2FmSWF4nKTA8K/3M8wWi7DuUv89DCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00c263e0dba94757c8e10cd7415a849ac203ac510fa422ed0d00f8089f1dd103","last_reissued_at":"2026-07-05T00:03:58.273261Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:03:58.273261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An Omega(n^2) Lower Bound for Random Universal Sets for Planar Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","math.CO"],"primary_cat":"cs.DM","authors_text":"Alexander Choi, Kevin Costello, Marek Chrobak","submitted_at":"2019-08-19T22:57:53Z","abstract_excerpt":"A set $U\\subseteq \\reals^2$ is $n$-universal if all $n$-vertex planar graphs have a planar straight-line embedding into $U$. We prove that if $Q \\subseteq \\reals^2$ consists of points chosen randomly and uniformly from the unit square then $Q$ must have cardinality $\\Omega(n^2)$ in order to be $n$-universal with high probability. This shows that the probabilistic method, at least in its basic form, cannot be used to establish an $o(n^2)$ upper bound on universal sets."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.07097","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.07097/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.07097","created_at":"2026-07-05T00:03:58.273322+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.07097v2","created_at":"2026-07-05T00:03:58.273322+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.07097","created_at":"2026-07-05T00:03:58.273322+00:00"},{"alias_kind":"pith_short_12","alias_value":"ADBGHYG3VFDV","created_at":"2026-07-05T00:03:58.273322+00:00"},{"alias_kind":"pith_short_16","alias_value":"ADBGHYG3VFDVPSHB","created_at":"2026-07-05T00:03:58.273322+00:00"},{"alias_kind":"pith_short_8","alias_value":"ADBGHYG3","created_at":"2026-07-05T00:03:58.273322+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/ADBGHYG3VFDVPSHBBTLUCWUETL","json":"https://pith.science/pith/ADBGHYG3VFDVPSHBBTLUCWUETL.json","graph_json":"https://pith.science/api/pith-number/ADBGHYG3VFDVPSHBBTLUCWUETL/graph.json","events_json":"https://pith.science/api/pith-number/ADBGHYG3VFDVPSHBBTLUCWUETL/events.json","paper":"https://pith.science/paper/ADBGHYG3"},"agent_actions":{"view_html":"https://pith.science/pith/ADBGHYG3VFDVPSHBBTLUCWUETL","download_json":"https://pith.science/pith/ADBGHYG3VFDVPSHBBTLUCWUETL.json","view_paper":"https://pith.science/paper/ADBGHYG3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.07097&json=true","fetch_graph":"https://pith.science/api/pith-number/ADBGHYG3VFDVPSHBBTLUCWUETL/graph.json","fetch_events":"https://pith.science/api/pith-number/ADBGHYG3VFDVPSHBBTLUCWUETL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ADBGHYG3VFDVPSHBBTLUCWUETL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ADBGHYG3VFDVPSHBBTLUCWUETL/action/storage_attestation","attest_author":"https://pith.science/pith/ADBGHYG3VFDVPSHBBTLUCWUETL/action/author_attestation","sign_citation":"https://pith.science/pith/ADBGHYG3VFDVPSHBBTLUCWUETL/action/citation_signature","submit_replication":"https://pith.science/pith/ADBGHYG3VFDVPSHBBTLUCWUETL/action/replication_record"}},"created_at":"2026-07-05T00:03:58.273322+00:00","updated_at":"2026-07-05T00:03:58.273322+00:00"}