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We show that if $\\chi(F)=\\ell>r \\geq 2$, then $ {\\rm ex}(n,H_F^{(r)})= t_r (n,\\ell-1)+ \\Theta( {\\rm biex}(n,F)n^{r-2}),$ where $t_r (n,\\ell-1)$ is the number of edges of an $n$-vertex complete balanced $\\ell-1$ partite $r$-graph and ${\\rm biex}(n,F)$ is the extremal number of the decomposition family of $F$. Since ${\\rm biex}(n,F)=O(n^{2-\\gamma})$ for some $\\gamma>0$, this improves on the bound ${\\rm ex}(n,H_F^{(r)})= t_r (n,\\ell-1)+ o(n^r)$ by Mubayi (2016). Furthermo"},"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":"1904.00146","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-30T04:22:36Z","cross_cats_sorted":[],"title_canon_sha256":"f1ab3df181e6fc58996c374d72cf2c7ca1ca5a2b4b5b7bcb2a276be9eeabc5a4","abstract_canon_sha256":"b1abfc837b21d059e3cd4f42cd8d191be725981567328d38fe7c487b6de8d07c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:49.310432Z","signature_b64":"hKonScjOJjg1YcHo3jabwL71CeV6UWeEmsNJykaMSpmls9QGxvDRLguN+BKrSgKx81G00oJ0XfUnEIHAVUmuCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f005e5dc13b474c4ed715dc0d93192e6b120cc2e1b2cba23a786f4fa372643d","last_reissued_at":"2026-05-17T23:49:49.309776Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:49.309776Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An Improved Error Term for Tur$\\acute{\\rm a}$n Number of Expanded Non-degenerate 2-graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guiying Yan, Xin Xu, Yucong Tang","submitted_at":"2019-03-30T04:22:36Z","abstract_excerpt":"For a 2-graph $F$, let $H_F^{(r)}$ be the $r$-graph obtained from $F$ by enlarging each edge with a new set of $r-2$ vertices. We show that if $\\chi(F)=\\ell>r \\geq 2$, then $ {\\rm ex}(n,H_F^{(r)})= t_r (n,\\ell-1)+ \\Theta( {\\rm biex}(n,F)n^{r-2}),$ where $t_r (n,\\ell-1)$ is the number of edges of an $n$-vertex complete balanced $\\ell-1$ partite $r$-graph and ${\\rm biex}(n,F)$ is the extremal number of the decomposition family of $F$. Since ${\\rm biex}(n,F)=O(n^{2-\\gamma})$ for some $\\gamma>0$, this improves on the bound ${\\rm ex}(n,H_F^{(r)})= t_r (n,\\ell-1)+ o(n^r)$ by Mubayi (2016). 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