{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:C2IZDSLS7T3JPO4U3LFEEM3D2K","short_pith_number":"pith:C2IZDSLS","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"},"canonical_sha256":"169191c972fcf697bb94daca423363d28d66821e8d47394f3f4d8f8ebf7608b5","source":{"kind":"arxiv","id":"2306.10375","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2306.10375","created_at":"2026-07-05T07:54:01Z"},{"alias_kind":"arxiv_version","alias_value":"2306.10375v3","created_at":"2026-07-05T07:54:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2306.10375","created_at":"2026-07-05T07:54:01Z"},{"alias_kind":"pith_short_12","alias_value":"C2IZDSLS7T3J","created_at":"2026-07-05T07:54:01Z"},{"alias_kind":"pith_short_16","alias_value":"C2IZDSLS7T3JPO4U","created_at":"2026-07-05T07:54:01Z"},{"alias_kind":"pith_short_8","alias_value":"C2IZDSLS","created_at":"2026-07-05T07:54:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:C2IZDSLS7T3JPO4U3LFEEM3D2K","target":"record","payload":{"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"},"canonical_sha256":"169191c972fcf697bb94daca423363d28d66821e8d47394f3f4d8f8ebf7608b5","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"},"source_kind":"arxiv","source_id":"2306.10375","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T07:54:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HgbJeZQNm1fn/d+fcYy5QzXDQNz01+f8tIO3/eiRC5wBLVWGw9BMHJt2BOIhwd3whBEUyVlPkEJg8KEIZIipAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T15:37:46.275304Z"},"content_sha256":"e148a1b2bcb2d7418f955f8211b51e5ef3b46cf1997043ae16739e4bea9d7cc7","schema_version":"1.0","event_id":"sha256:e148a1b2bcb2d7418f955f8211b51e5ef3b46cf1997043ae16739e4bea9d7cc7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:C2IZDSLS7T3JPO4U3LFEEM3D2K","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T07:54:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q23MGlTcmS9engYaWXZGInLSdi6eittl/JPIXAotoWPG11PJvlKdLVJcE9Wcy/AfxeLCHCDTNplOFSqoXC0XBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T15:37:46.275676Z"},"content_sha256":"2ef231c59e84baa018baf332756e78abc9481e7d5c4aaaf44aaafb34836839ab","schema_version":"1.0","event_id":"sha256:2ef231c59e84baa018baf332756e78abc9481e7d5c4aaaf44aaafb34836839ab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K/bundle.json","state_url":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-07T15:37:46Z","links":{"resolver":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K","bundle":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K/bundle.json","state":"https://pith.science/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/C2IZDSLS7T3JPO4U3LFEEM3D2K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:C2IZDSLS7T3JPO4U3LFEEM3D2K","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"280bc9ee52a78b9ab8f61b2cd0777c62aab34926077005ce3ba442a403f2ddde","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2023-06-17T15:23:13Z","title_canon_sha256":"f938ad616a718b3d3ac650437f2eb736cb6de2e1407ccd47a684e3d1f44a6499"},"schema_version":"1.0","source":{"id":"2306.10375","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2306.10375","created_at":"2026-07-05T07:54:01Z"},{"alias_kind":"arxiv_version","alias_value":"2306.10375v3","created_at":"2026-07-05T07:54:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2306.10375","created_at":"2026-07-05T07:54:01Z"},{"alias_kind":"pith_short_12","alias_value":"C2IZDSLS7T3J","created_at":"2026-07-05T07:54:01Z"},{"alias_kind":"pith_short_16","alias_value":"C2IZDSLS7T3JPO4U","created_at":"2026-07-05T07:54:01Z"},{"alias_kind":"pith_short_8","alias_value":"C2IZDSLS","created_at":"2026-07-05T07:54:01Z"}],"graph_snapshots":[{"event_id":"sha256:2ef231c59e84baa018baf332756e78abc9481e7d5c4aaaf44aaafb34836839ab","target":"graph","created_at":"2026-07-05T07:54:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2306.10375/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Ali Mohammadian, Behruz Tayfeh-Rezaie, Meysam Miralaei, Olga Kalinichenko","cross_cats":[],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2023-06-17T15:23:13Z","title":"Weak saturation numbers in random graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.10375","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e148a1b2bcb2d7418f955f8211b51e5ef3b46cf1997043ae16739e4bea9d7cc7","target":"record","created_at":"2026-07-05T07:54:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"280bc9ee52a78b9ab8f61b2cd0777c62aab34926077005ce3ba442a403f2ddde","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2023-06-17T15:23:13Z","title_canon_sha256":"f938ad616a718b3d3ac650437f2eb736cb6de2e1407ccd47a684e3d1f44a6499"},"schema_version":"1.0","source":{"id":"2306.10375","kind":"arxiv","version":3}},"canonical_sha256":"169191c972fcf697bb94daca423363d28d66821e8d47394f3f4d8f8ebf7608b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"169191c972fcf697bb94daca423363d28d66821e8d47394f3f4d8f8ebf7608b5","first_computed_at":"2026-07-05T07:54:01.716648Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:54:01.716648Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DndN2q398SJMe5XFoot+yr7n1eN36NzsAyaUQTyATfKMDeicr3HbdVK49bdHeBXn2P9pAlDimczijZNKRyw4DQ==","signature_status":"signed_v1","signed_at":"2026-07-05T07:54:01.717162Z","signed_message":"canonical_sha256_bytes"},"source_id":"2306.10375","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e148a1b2bcb2d7418f955f8211b51e5ef3b46cf1997043ae16739e4bea9d7cc7","sha256:2ef231c59e84baa018baf332756e78abc9481e7d5c4aaaf44aaafb34836839ab"],"state_sha256":"7a051e8a1a6a691dec7267179dd62099d8d583b65d04a34369f9327ca183eb81"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8ql1in3/9bHZEadaWTDnkxywtsXedqmW6PAjourAgPTMQxqBaMkoLVQIV5R0ZQLAJTDW8xvHDO9xaopN5wF7Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T15:37:46.278691Z","bundle_sha256":"277c3dc5749f3c8e50220ea6da12317758a47de6d2ebc9e347f87a865de936c6"}}