{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:B64WOLFKLPEFCHTANVD6WSDLNR","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":"d6ade47717251fead72fc61143e1810865d8ec5756ba58b0898e8621ff8046b0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-27T11:59:12Z","title_canon_sha256":"6957e5fe99601299df2f503d5a267e7aef328d5c13a79f604fb35f99b64e5f3d"},"schema_version":"1.0","source":{"id":"1806.10425","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.10425","created_at":"2026-05-18T00:12:12Z"},{"alias_kind":"arxiv_version","alias_value":"1806.10425v1","created_at":"2026-05-18T00:12:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.10425","created_at":"2026-05-18T00:12:12Z"},{"alias_kind":"pith_short_12","alias_value":"B64WOLFKLPEF","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"B64WOLFKLPEFCHTA","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"B64WOLFK","created_at":"2026-05-18T12:32:13Z"}],"graph_snapshots":[{"event_id":"sha256:99455bdd25cb5f3ba9b714a0d274b29b421f4eba9ce092a4f2cd78a525a1b0b7","target":"graph","created_at":"2026-05-18T00:12:12Z","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"},"paper":{"abstract_excerpt":"Given two graphs $G$ and $H$, it is said that $G$ percolates in $H$-bootstrap process if one could join all the nonadjacent pairs of vertices of $G$ in some order such that a new copy of $H$ is created at each step. Balogh, Bollob\\'as and Morris in 2012 investigated the threshold of $H$-bootstrap percolation in the Erd\\H{o}s-R\\'enyi model for the complete graph $H$ and proposed the similar problem for $H=K_{s,t}$, the complete bipartite graph. In this paper, we provide lower and upper bounds on the threshold of $K_{2, t}$-bootstrap percolation. In addition, a threshold function is derived for ","authors_text":"A. Mohammadian, B. Tayfeh-Rezaie, M.R. Bidgoli","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-27T11:59:12Z","title":"On $K_{2,t}$-bootstrap percolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.10425","kind":"arxiv","version":1},"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":"sha256:218bdc2e761deeda94a1eb39e0fb11847b619d0f9e2029cd1a62649b9b925c90","target":"record","created_at":"2026-05-18T00:12:12Z","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":"d6ade47717251fead72fc61143e1810865d8ec5756ba58b0898e8621ff8046b0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-27T11:59:12Z","title_canon_sha256":"6957e5fe99601299df2f503d5a267e7aef328d5c13a79f604fb35f99b64e5f3d"},"schema_version":"1.0","source":{"id":"1806.10425","kind":"arxiv","version":1}},"canonical_sha256":"0fb9672caa5bc8511e606d47eb486b6c6daa7fd2c1a194ae778ddc540a0dd05e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0fb9672caa5bc8511e606d47eb486b6c6daa7fd2c1a194ae778ddc540a0dd05e","first_computed_at":"2026-05-18T00:12:12.483921Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:12.483921Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jYGd10AZePoXp8oZdD9dSXtb7znOAo2bEIGfoYVz3ZB2ErIuzkj6heQIj7/EUcv83PxPdAV3RKfvZP5mVhD7Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:12.484582Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.10425","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:218bdc2e761deeda94a1eb39e0fb11847b619d0f9e2029cd1a62649b9b925c90","sha256:99455bdd25cb5f3ba9b714a0d274b29b421f4eba9ce092a4f2cd78a525a1b0b7"],"state_sha256":"a7aa64b4add11f4c3e69e511f440c12fd15acd883f4269e4bfdd607cf5da9ef9"}