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Suppose that $\\cH$ has no $2$-cycles (regardless of sizes of edges), and neither contains $3$-cycles nor $4$-cycles consisting of $2$-element edges. If the average degrees $t_i^{i-1} := i |\\cE_i|/ |V|$ satisfy that $t_i^{i-1} \\leq T^{i-1} (\\ln T)^{\\frac{k-i}{k-1}}$ for $i= 2, \\dots , k$, then there exists a constant $C_k > 0$, depending only on $"},"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":"1602.03569","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-10T23:13:22Z","cross_cats_sorted":[],"title_canon_sha256":"cb1d9adeb671280a10689c3b32237b668821073e5b3b26be78d61d69f3cd7e52","abstract_canon_sha256":"5d2df8dccc5985fc8fbea3bdfceafec2a438858e76485aa2294c19b11723bbd4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:58.466515Z","signature_b64":"hLs3kBcJriSDZ7QtiVCDSUyBmJOkfc64raqP2LRe6FpBHm6h0HLQgb4OcDRs7NnjeSMnlnNg1SjMSCZDSntkCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8b785fecd28869da0f33040fe4e9859f39cfffeba84bb74712ce8ebb5520bc2","last_reissued_at":"2026-05-18T01:20:58.466083Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:58.466083Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The independence number of non-uniform uncrowded hypergraphs and an anti-Ramsey type result","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hanno Lefmann, Sang June Lee","submitted_at":"2016-02-10T23:13:22Z","abstract_excerpt":"We prove the following: Fix an integer $k\\geq 2$, and let $T$ be a real number with $T\\geq 1.5$. 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