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Recently this was improved to $\\frac{14}{15}(1+o(1)){n \\choose \\lfloor{\\frac{r}{2}}\\rfloor}$ for even $r \\geq 4$. A bound of $\\bigg[\\frac{r}{2}(\\frac{14}{15})^{\\frac{r}{4}}+o(1)\\bigg](1+o(1)){n \\choose \\lfloor{\\frac{r}{2}}\\rfloor}$ was also proved recently. The smallest odd $r$ for which $c_r < 1$ 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":"1712.06403","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-18T13:56:30Z","cross_cats_sorted":[],"title_canon_sha256":"9d028da81e84a38e0fd23c29156e91e6ecf0afb3f5cbd82651e918b551d24323","abstract_canon_sha256":"87e8d5f1daadf6b2ed5ea6b3978962bde9d563f0917b78da8f6623feb6f1fb53"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:35.549252Z","signature_b64":"V2kJF6L000I8RExUoARnM0ywyUqIZydiZIMPcXuYH/WGrI7hk3rdRUiyUugged9BZOsNH5Vg9kkyzxC2Y89zAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5bb240389d1ceafbcae1854f029d7053f0cb1fe40b08f8f4ed753152829abdf","last_reissued_at":"2026-05-18T00:27:35.548682Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:35.548682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounds for the Graham-Pollak Theorem for Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anand Babu, Sundar Vishwanathan","submitted_at":"2017-12-18T13:56:30Z","abstract_excerpt":"Let $f_r(n)$ represent the minimum number of complete $r$-partite $r$-graphs required to partition the edge set of the complete $r$-uniform hypergraph on $n$ vertices. 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