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This family is called an $r$-star. Then $G$ is said to be $r$-EKR if no intersecting subfamily of $ \\mathcal{I}^{(r)}(G)$ is bigger than the largest $r$-star. Let $k, n, r \\geq 1$, and let $T(n, k)$ be the tree of depth two in which the root has degree $n$ and every neighbou"},"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":"1811.04902","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-12T18:41:49Z","cross_cats_sorted":[],"title_canon_sha256":"b8b1cd93d6115f6118a91c27ddb2410edc327b0512277204b38366c7429139b4","abstract_canon_sha256":"0068ef1b6e33928ca57d8b6b5feffcf0e1c5e38b02c48f29d5504bda0e79acfd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:55.705166Z","signature_b64":"5yYTSnwQiDl47JFTXFfka58s0TIPTNLGm/ImkBhbt5xnQX5cmaHXm1guV7APWJVDOLcY++o124EKbrvK3kiSCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd1cac26f3a75f36e7168b6759d7b9f8d49143d8b78ea560d274c293d424de49","last_reissued_at":"2026-05-17T23:40:55.704545Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:55.704545Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Erd\\H{o}s-Ko-Rado property of trees of depth two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carl Feghali","submitted_at":"2018-11-12T18:41:49Z","abstract_excerpt":"A family of sets is intersecting if any two sets in the family intersect. Given a graph $G$ and an integer $r\\geq 1$, let $\\mathcal{I}^{(r)}(G)$ denote the family of independent sets of size $r$ of $G$. For a vertex $v$ of $G$, let $\\mathcal{I}^{(r)}_v(G)$ denote the family of independent sets of size $r$ that contain $v$. This family is called an $r$-star. Then $G$ is said to be $r$-EKR if no intersecting subfamily of $ \\mathcal{I}^{(r)}(G)$ is bigger than the largest $r$-star. Let $k, n, r \\geq 1$, and let $T(n, k)$ be the tree of depth two in which the root has degree $n$ and every neighbou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.04902","kind":"arxiv","version":6},"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":"1811.04902","created_at":"2026-05-17T23:40:55.704627+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.04902v6","created_at":"2026-05-17T23:40:55.704627+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.04902","created_at":"2026-05-17T23:40:55.704627+00:00"},{"alias_kind":"pith_short_12","alias_value":"7UOKYJXTU5PT","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_16","alias_value":"7UOKYJXTU5PTNZYW","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_8","alias_value":"7UOKYJXT","created_at":"2026-05-18T12:32:11.075285+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/7UOKYJXTU5PTNZYWRNTVTV5Z7D","json":"https://pith.science/pith/7UOKYJXTU5PTNZYWRNTVTV5Z7D.json","graph_json":"https://pith.science/api/pith-number/7UOKYJXTU5PTNZYWRNTVTV5Z7D/graph.json","events_json":"https://pith.science/api/pith-number/7UOKYJXTU5PTNZYWRNTVTV5Z7D/events.json","paper":"https://pith.science/paper/7UOKYJXT"},"agent_actions":{"view_html":"https://pith.science/pith/7UOKYJXTU5PTNZYWRNTVTV5Z7D","download_json":"https://pith.science/pith/7UOKYJXTU5PTNZYWRNTVTV5Z7D.json","view_paper":"https://pith.science/paper/7UOKYJXT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.04902&json=true","fetch_graph":"https://pith.science/api/pith-number/7UOKYJXTU5PTNZYWRNTVTV5Z7D/graph.json","fetch_events":"https://pith.science/api/pith-number/7UOKYJXTU5PTNZYWRNTVTV5Z7D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7UOKYJXTU5PTNZYWRNTVTV5Z7D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7UOKYJXTU5PTNZYWRNTVTV5Z7D/action/storage_attestation","attest_author":"https://pith.science/pith/7UOKYJXTU5PTNZYWRNTVTV5Z7D/action/author_attestation","sign_citation":"https://pith.science/pith/7UOKYJXTU5PTNZYWRNTVTV5Z7D/action/citation_signature","submit_replication":"https://pith.science/pith/7UOKYJXTU5PTNZYWRNTVTV5Z7D/action/replication_record"}},"created_at":"2026-05-17T23:40:55.704627+00:00","updated_at":"2026-05-17T23:40:55.704627+00:00"}