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Improving recent results of Ferrara, Jacobson, Pfender and Wenger, and generalizing a recent result of Roberts, we define a function $\\alpha(k,r)$ such that $sat(n,k,r) = \\alpha(k,r)n + o(n)$ as $n \\rightarrow \\infty$. Moreover, we prove that \\[ k(2r-4) \\le \\alpha(k,r) \\le \\begin{cases} (k-1)(4r-k-6) &\\text{ for }r \\le k \\le 2r-3,"},"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":"1708.01607","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-04T17:58:30Z","cross_cats_sorted":[],"title_canon_sha256":"a0846ff807c8ef96e29f8c7a53ff84334180524cb5a48bcf716fe5edf080a053","abstract_canon_sha256":"f376f937eb783300708838090bfdff00f6118d5cade5be1e34c9290db07b87e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:00.991479Z","signature_b64":"EGVdRiBfSlZgSIKX6zXyA/pCy0PzFNvv30eO2iPTLNjIAZ/ejcoK3mksFAm/TQub/6TC4W8h6DjyvkdkeWsZDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2fb648afb1c6116bb2a4873ee68ba4aace1a5be1c00d3fc24efe6146397b9584","last_reissued_at":"2026-05-18T00:32:00.991022Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:00.991022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Partite Saturation of Complete Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ant\\'onio Gir\\~ao, Kamil Popielarz, Teeradej Kittipassorn","submitted_at":"2017-08-04T17:58:30Z","abstract_excerpt":"We study the problem of determining $sat(n,k,r)$, the minimum number of edges in a $k$-partite graph $G$ with $n$ vertices in each part such that $G$ is $K_r$-free but the addition of an edge joining any two non-adjacent vertices from different parts creates a $K_r$. 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