{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:6FZMB36WMYEKH7OQUM6UE462M2","short_pith_number":"pith:6FZMB36W","schema_version":"1.0","canonical_sha256":"f172c0efd66608a3fdd0a33d4273da66a3ba4e8bc0cd496b8234603beef9b529","source":{"kind":"arxiv","id":"1504.05324","version":1},"attestation_state":"computed","paper":{"title":"Random Geometric Graphs and Isometries of Normed Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.FA","authors_text":"B\\'ela Bollob\\'as, Imre Leader, Karen Gunderson, Mark Walters, Paul Balister","submitted_at":"2015-04-21T07:45:26Z","abstract_excerpt":"Given a countable dense subset $S$ of a finite-dimensional normed space $X$, and $0<p<1$, we form a random graph on $S$ by joining, independently and with probability $p$, each pair of points at distance less than $1$. We say that $S$ is `Rado' if any two such random graphs are (almost surely) isomorphic.\n  Bonato and Janssen showed that in $l_\\infty^d$ almost all $S$ are Rado. Our main aim in this paper is to show that $l_\\infty^d$ is the unique normed space with this property: indeed, in every other space almost all sets $S$ are non-Rado. We also determine which spaces admit some Rado set: t"},"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":"1504.05324","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-04-21T07:45:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"cbde2ab53b14c52e4da37f7c31a6315c28f608efb699f738e6bef0399e06f5de","abstract_canon_sha256":"381b9f1ec1477af328eb7ac55b99be5e4bb840c3c082a9b9d6ae79c08cb70866"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:14.535744Z","signature_b64":"eoe/qYhpR5vGbFrlM4yfW38Uw4B9bUByAEAiQLgvYUBXb8vBDrpRH4aPsrac9jH4bmyHp2u1cTNgtYTRUfOgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f172c0efd66608a3fdd0a33d4273da66a3ba4e8bc0cd496b8234603beef9b529","last_reissued_at":"2026-05-18T02:18:14.535116Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:14.535116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random Geometric Graphs and Isometries of Normed Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.FA","authors_text":"B\\'ela Bollob\\'as, Imre Leader, Karen Gunderson, Mark Walters, Paul Balister","submitted_at":"2015-04-21T07:45:26Z","abstract_excerpt":"Given a countable dense subset $S$ of a finite-dimensional normed space $X$, and $0<p<1$, we form a random graph on $S$ by joining, independently and with probability $p$, each pair of points at distance less than $1$. We say that $S$ is `Rado' if any two such random graphs are (almost surely) isomorphic.\n  Bonato and Janssen showed that in $l_\\infty^d$ almost all $S$ are Rado. Our main aim in this paper is to show that $l_\\infty^d$ is the unique normed space with this property: indeed, in every other space almost all sets $S$ are non-Rado. We also determine which spaces admit some Rado set: t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.05324","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":"1504.05324","created_at":"2026-05-18T02:18:14.535204+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.05324v1","created_at":"2026-05-18T02:18:14.535204+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.05324","created_at":"2026-05-18T02:18:14.535204+00:00"},{"alias_kind":"pith_short_12","alias_value":"6FZMB36WMYEK","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_16","alias_value":"6FZMB36WMYEKH7OQ","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_8","alias_value":"6FZMB36W","created_at":"2026-05-18T12:29:07.941421+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/6FZMB36WMYEKH7OQUM6UE462M2","json":"https://pith.science/pith/6FZMB36WMYEKH7OQUM6UE462M2.json","graph_json":"https://pith.science/api/pith-number/6FZMB36WMYEKH7OQUM6UE462M2/graph.json","events_json":"https://pith.science/api/pith-number/6FZMB36WMYEKH7OQUM6UE462M2/events.json","paper":"https://pith.science/paper/6FZMB36W"},"agent_actions":{"view_html":"https://pith.science/pith/6FZMB36WMYEKH7OQUM6UE462M2","download_json":"https://pith.science/pith/6FZMB36WMYEKH7OQUM6UE462M2.json","view_paper":"https://pith.science/paper/6FZMB36W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.05324&json=true","fetch_graph":"https://pith.science/api/pith-number/6FZMB36WMYEKH7OQUM6UE462M2/graph.json","fetch_events":"https://pith.science/api/pith-number/6FZMB36WMYEKH7OQUM6UE462M2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6FZMB36WMYEKH7OQUM6UE462M2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6FZMB36WMYEKH7OQUM6UE462M2/action/storage_attestation","attest_author":"https://pith.science/pith/6FZMB36WMYEKH7OQUM6UE462M2/action/author_attestation","sign_citation":"https://pith.science/pith/6FZMB36WMYEKH7OQUM6UE462M2/action/citation_signature","submit_replication":"https://pith.science/pith/6FZMB36WMYEKH7OQUM6UE462M2/action/replication_record"}},"created_at":"2026-05-18T02:18:14.535204+00:00","updated_at":"2026-05-18T02:18:14.535204+00:00"}