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Let the minimum $\\ell$-degree of $H_n$ be $\\delta_{\\ell}(H_n) = \\min \\{ \\deg(T) : T \\subseteq V(H_n)$ and $|T|=\\ell\\}$. Given a family $\\mathcal{F}$ of $k$-graphs, the $\\ell$-degree Tur\\'an number $\\text{ex}_{\\ell}(n, \\mathcal{F})$ is the largest $\\delta_{\\ell}(H_n)$ over all $\\mathcal{F}$-free $k$-graphs $H_n$ on $n$ vertices. Hence, $\\text{ex}_0(n, \\mathcal{F})$ is the Tur\\'a"},"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":"1210.5726","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-21T13:18:01Z","cross_cats_sorted":[],"title_canon_sha256":"72e6d6bb2e4d38eefb25c8775f3e68b5e46c50fcaf971f8eb8fec873b92923cd","abstract_canon_sha256":"fa7af678aaee317b2dc35df1382726819514e7987ecf55fd96c12fb387810099"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:40:11.077705Z","signature_b64":"mt/KsasdSCjz44EH671ts/KP+ziQ3p5cywLPfhTMx3LhP1CA75trd7QdpZZkUPyoPbFGcKgg8Tz+sH8wgKsDBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0a4db73429f2fa6b613843fbe48c1fe921e5ad45e083e35bb2f502cd797a236","last_reissued_at":"2026-05-18T02:40:11.077076Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:40:11.077076Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$\\ell$-degree Tur\\'an density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Allan Lo, Klas Markstr\\\"om","submitted_at":"2012-10-21T13:18:01Z","abstract_excerpt":"Let $H_n$ be a $k$-graph on $n$ vertices. For $0 \\le \\ell <k$ and an $\\ell$-subset $T$ of $V(H_n)$, define the degree $\\deg(T)$ of $T$ to be the number of $(k-\\ell)$-subsets~$S$ such that $S \\cup T$ is an edge in~$H_n$. Let the minimum $\\ell$-degree of $H_n$ be $\\delta_{\\ell}(H_n) = \\min \\{ \\deg(T) : T \\subseteq V(H_n)$ and $|T|=\\ell\\}$. Given a family $\\mathcal{F}$ of $k$-graphs, the $\\ell$-degree Tur\\'an number $\\text{ex}_{\\ell}(n, \\mathcal{F})$ is the largest $\\delta_{\\ell}(H_n)$ over all $\\mathcal{F}$-free $k$-graphs $H_n$ on $n$ vertices. Hence, $\\text{ex}_0(n, \\mathcal{F})$ is the Tur\\'a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5726","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":"1210.5726","created_at":"2026-05-18T02:40:11.077164+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.5726v2","created_at":"2026-05-18T02:40:11.077164+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5726","created_at":"2026-05-18T02:40:11.077164+00:00"},{"alias_kind":"pith_short_12","alias_value":"4CSNW42CT4X2","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"4CSNW42CT4X2NNQT","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"4CSNW42C","created_at":"2026-05-18T12:26:53.410803+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2607.06518","citing_title":"Tree suspensions and transfer functions for single degree Tur\\'an spectra","ref_index":24,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4CSNW42CT4X2NNQTQQ734SGB72","json":"https://pith.science/pith/4CSNW42CT4X2NNQTQQ734SGB72.json","graph_json":"https://pith.science/api/pith-number/4CSNW42CT4X2NNQTQQ734SGB72/graph.json","events_json":"https://pith.science/api/pith-number/4CSNW42CT4X2NNQTQQ734SGB72/events.json","paper":"https://pith.science/paper/4CSNW42C"},"agent_actions":{"view_html":"https://pith.science/pith/4CSNW42CT4X2NNQTQQ734SGB72","download_json":"https://pith.science/pith/4CSNW42CT4X2NNQTQQ734SGB72.json","view_paper":"https://pith.science/paper/4CSNW42C","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.5726&json=true","fetch_graph":"https://pith.science/api/pith-number/4CSNW42CT4X2NNQTQQ734SGB72/graph.json","fetch_events":"https://pith.science/api/pith-number/4CSNW42CT4X2NNQTQQ734SGB72/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4CSNW42CT4X2NNQTQQ734SGB72/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4CSNW42CT4X2NNQTQQ734SGB72/action/storage_attestation","attest_author":"https://pith.science/pith/4CSNW42CT4X2NNQTQQ734SGB72/action/author_attestation","sign_citation":"https://pith.science/pith/4CSNW42CT4X2NNQTQQ734SGB72/action/citation_signature","submit_replication":"https://pith.science/pith/4CSNW42CT4X2NNQTQQ734SGB72/action/replication_record"}},"created_at":"2026-05-18T02:40:11.077164+00:00","updated_at":"2026-05-18T02:40:11.077164+00:00"}