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The weak saturation number $ wsat(G, F)$ is the minimum number of edges of a weakly $(G, F)$-saturated graph. In this paper, we deal with the relation between $ wsat(G(n,p), F)$ and $ wsat(K_n, F)$, where $G(n,p)$ denotes the Erd\\H{o}s--R\\'enyi random graph and $ K_n$ denotes the complete graph on $ n$ vertic"},"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":"2306.10375","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2023-06-17T15:23:13Z","cross_cats_sorted":[],"title_canon_sha256":"f938ad616a718b3d3ac650437f2eb736cb6de2e1407ccd47a684e3d1f44a6499","abstract_canon_sha256":"280bc9ee52a78b9ab8f61b2cd0777c62aab34926077005ce3ba442a403f2ddde"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:54:01.717162Z","signature_b64":"DndN2q398SJMe5XFoot+yr7n1eN36NzsAyaUQTyATfKMDeicr3HbdVK49bdHeBXn2P9pAlDimczijZNKRyw4DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"169191c972fcf697bb94daca423363d28d66821e8d47394f3f4d8f8ebf7608b5","last_reissued_at":"2026-07-05T07:54:01.716648Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:54:01.716648Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak saturation numbers in random graphs","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ali Mohammadian, Behruz Tayfeh-Rezaie, Meysam Miralaei, Olga Kalinichenko","submitted_at":"2023-06-17T15:23:13Z","abstract_excerpt":"For two given graphs $G$ and $F$, a graph $ H$ is said to be weakly $ (G, F) $-saturated if $H$ is a spanning subgraph of $ G$ which has no copy of $F$ as a subgraph and one can add all edges in $ E(G)\\setminus E(H)$ to $ H$ in some order so that a new copy of $F$ is created at each step. The weak saturation number $ wsat(G, F)$ is the minimum number of edges of a weakly $(G, F)$-saturated graph. In this paper, we deal with the relation between $ wsat(G(n,p), F)$ and $ wsat(K_n, F)$, where $G(n,p)$ denotes the Erd\\H{o}s--R\\'enyi random graph and $ K_n$ denotes the complete graph on $ n$ vertic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.10375","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2306.10375/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2306.10375","created_at":"2026-07-05T07:54:01.716716+00:00"},{"alias_kind":"arxiv_version","alias_value":"2306.10375v3","created_at":"2026-07-05T07:54:01.716716+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2306.10375","created_at":"2026-07-05T07:54:01.716716+00:00"},{"alias_kind":"pith_short_12","alias_value":"C2IZDSLS7T3J","created_at":"2026-07-05T07:54:01.716716+00:00"},{"alias_kind":"pith_short_16","alias_value":"C2IZDSLS7T3JPO4U","created_at":"2026-07-05T07:54:01.716716+00:00"},{"alias_kind":"pith_short_8","alias_value":"C2IZDSLS","created_at":"2026-07-05T07:54:01.716716+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.05697","citing_title":"When does a tree activate the random graph?","ref_index":42,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K","json":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K.json","graph_json":"https://pith.science/api/pith-number/C2IZDSLS7T3JPO4U3LFEEM3D2K/graph.json","events_json":"https://pith.science/api/pith-number/C2IZDSLS7T3JPO4U3LFEEM3D2K/events.json","paper":"https://pith.science/paper/C2IZDSLS"},"agent_actions":{"view_html":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K","download_json":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K.json","view_paper":"https://pith.science/paper/C2IZDSLS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2306.10375&json=true","fetch_graph":"https://pith.science/api/pith-number/C2IZDSLS7T3JPO4U3LFEEM3D2K/graph.json","fetch_events":"https://pith.science/api/pith-number/C2IZDSLS7T3JPO4U3LFEEM3D2K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K/action/storage_attestation","attest_author":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K/action/author_attestation","sign_citation":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K/action/citation_signature","submit_replication":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K/action/replication_record"}},"created_at":"2026-07-05T07:54:01.716716+00:00","updated_at":"2026-07-05T07:54:01.716716+00:00"}