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The $k$-degree of the set $X$, denoted ${\\rm deg}_k(X)$, is defined as the maximum degree over all $k$-element subset of $X$.\n  We show that if $X$ is a random point set consisting of $n$ independently and uniformly chosen points from a compact set $K$ then ${\\rm deg}_d(X)=\\Th"},"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":"1808.08734","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-08-27T08:39:45Z","cross_cats_sorted":[],"title_canon_sha256":"e7ee2d6bacb8f0f9e170be122d8700e8d1976f652d8d420e604b74234a08bca2","abstract_canon_sha256":"c5ac1bf6c7f0301d498840cdcb7329483064a3c804ae2dbac0187d11b1d4c413"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:02.923031Z","signature_b64":"ArFciJS9j4Guo3IHxUuEW3ZcElLUPYGzF3pmSE/FzmlF+vmk4pwRTtHUnLom0ltotCmjadVSeChAN58Zz4R8Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3aeec11c8f0e2a7b46f9ef48896562c2c4dd246c95d8a31ffee6e6b753cee988","last_reissued_at":"2026-05-17T23:47:02.922397Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:02.922397Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stars of Empty Simplices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniel Temesvari, Matthias Reitzner","submitted_at":"2018-08-27T08:39:45Z","abstract_excerpt":"Let $X=\\{x_1,\\ldots,x_n\\} \\subset \\mathbb R^d$ be an $n$-element point set in general position. 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The $k$-degree of the set $X$, denoted ${\\rm deg}_k(X)$, is defined as the maximum degree over all $k$-element subset of $X$.\n  We show that if $X$ is a random point set consisting of $n$ independently and uniformly chosen points from a compact set $K$ then ${\\rm deg}_d(X)=\\Th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08734","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":""},"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":"1808.08734","created_at":"2026-05-17T23:47:02.922519+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.08734v2","created_at":"2026-05-17T23:47:02.922519+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.08734","created_at":"2026-05-17T23:47:02.922519+00:00"},{"alias_kind":"pith_short_12","alias_value":"HLXMCHEPBYVH","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"HLXMCHEPBYVHWRXZ","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"HLXMCHEP","created_at":"2026-05-18T12:32:28.185984+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/HLXMCHEPBYVHWRXZ55EISZLCYL","json":"https://pith.science/pith/HLXMCHEPBYVHWRXZ55EISZLCYL.json","graph_json":"https://pith.science/api/pith-number/HLXMCHEPBYVHWRXZ55EISZLCYL/graph.json","events_json":"https://pith.science/api/pith-number/HLXMCHEPBYVHWRXZ55EISZLCYL/events.json","paper":"https://pith.science/paper/HLXMCHEP"},"agent_actions":{"view_html":"https://pith.science/pith/HLXMCHEPBYVHWRXZ55EISZLCYL","download_json":"https://pith.science/pith/HLXMCHEPBYVHWRXZ55EISZLCYL.json","view_paper":"https://pith.science/paper/HLXMCHEP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.08734&json=true","fetch_graph":"https://pith.science/api/pith-number/HLXMCHEPBYVHWRXZ55EISZLCYL/graph.json","fetch_events":"https://pith.science/api/pith-number/HLXMCHEPBYVHWRXZ55EISZLCYL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HLXMCHEPBYVHWRXZ55EISZLCYL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HLXMCHEPBYVHWRXZ55EISZLCYL/action/storage_attestation","attest_author":"https://pith.science/pith/HLXMCHEPBYVHWRXZ55EISZLCYL/action/author_attestation","sign_citation":"https://pith.science/pith/HLXMCHEPBYVHWRXZ55EISZLCYL/action/citation_signature","submit_replication":"https://pith.science/pith/HLXMCHEPBYVHWRXZ55EISZLCYL/action/replication_record"}},"created_at":"2026-05-17T23:47:02.922519+00:00","updated_at":"2026-05-17T23:47:02.922519+00:00"}