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In this paper we investigate the semi-saturation number of $K_r$ on $n$ vertices with maximal degree at most $\\Delta$, denoted by $ssat^{\\Delta}(n,K_r)$. This investigation was suggested by Erd\\H os, R\\'enyi and S\\'os, who in 1966 considered the graph of diameter 2 with degree restrictions, equivalently $ssat^{\\Delta}(n,K_3)$.\n  The following are s"},"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":"2606.28727","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-27T04:17:44Z","cross_cats_sorted":[],"title_canon_sha256":"74bfaf2ffc6a37cc679cc047052e0c8242f5eabd5e66fa5e36a674719e21e2d4","abstract_canon_sha256":"4b2b2f3e63c55f3652278216efada4062c145701d80933b7163c6508a80d2f74"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T01:16:49.538167Z","signature_b64":"RpeSyuo8zZLWmVRwZC0qHK20qEVgZ0JKSO+O79pMKRuLmNmxbU2kCjwxSN9tily6WAF5t41nLp/RgvAwtWjgAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93ca17213137c5700075f378d2ce169cdee944a1b8f0050821789c4e0576e130","last_reissued_at":"2026-06-30T01:16:49.537652Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T01:16:49.537652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Degree-restricted semi-saturation numbers of cliques and its applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mei Lu, Yanzhe Qiu, Yiduo Xu, Zhen He","submitted_at":"2026-06-27T04:17:44Z","abstract_excerpt":"A graph $G$ is said to be $F$-semi-saturated if the addition of any nonedge $e \\not \\in E(G)$ would create a new copy of $F$ in $G+e$. 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