{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:E2ZNQQ4TZKWPMKXR3H3MWNQ2GV","short_pith_number":"pith:E2ZNQQ4T","schema_version":"1.0","canonical_sha256":"26b2d84393caacf62af1d9f6cb361a3573933b5c17e961079e58862f2174a733","source":{"kind":"arxiv","id":"1203.4223","version":3},"attestation_state":"computed","paper":{"title":"Random triangle removal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Alan Frieze, Eyal Lubetzky, Tom Bohman","submitted_at":"2012-03-19T19:58:19Z","abstract_excerpt":"Starting from a complete graph on $n$ vertices, repeatedly delete the edges of a uniformly chosen triangle. This stochastic process terminates once it arrives at a triangle-free graph, and the fundamental question is to estimate the final number of edges (equivalently, the time it takes the process to finish, or how many edge-disjoint triangles are packed via the random greedy algorithm). Bollob\\'as and Erd\\H{o}s (1990) conjectured that the expected final number of edges has order $n^{3/2}$, motivated by the study of the Ramsey number $R(3,t)$. An upper bound of $o(n^2)$ was shown by Spencer ("},"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":"1203.4223","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-19T19:58:19Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"72a81e864f2cae1ca2d78f7d6d175e51383942dd60a94b7bd1896bf605a91f69","abstract_canon_sha256":"8a6d65aee35da995c8519e470a777172bc1f775124f09662437393128ba549e0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:04.027642Z","signature_b64":"7Z4Q/j9TKC3ZmrGpqwGTVu9vPadg0R0QWZ7SZ12OSxASe5hA0oDrbc7RMjcTpesx0F5igm4I28duLqcVejvGAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"26b2d84393caacf62af1d9f6cb361a3573933b5c17e961079e58862f2174a733","last_reissued_at":"2026-05-18T03:54:04.027142Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:04.027142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random triangle removal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Alan Frieze, Eyal Lubetzky, Tom Bohman","submitted_at":"2012-03-19T19:58:19Z","abstract_excerpt":"Starting from a complete graph on $n$ vertices, repeatedly delete the edges of a uniformly chosen triangle. This stochastic process terminates once it arrives at a triangle-free graph, and the fundamental question is to estimate the final number of edges (equivalently, the time it takes the process to finish, or how many edge-disjoint triangles are packed via the random greedy algorithm). Bollob\\'as and Erd\\H{o}s (1990) conjectured that the expected final number of edges has order $n^{3/2}$, motivated by the study of the Ramsey number $R(3,t)$. An upper bound of $o(n^2)$ was shown by Spencer ("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.4223","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":""},"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":"1203.4223","created_at":"2026-05-18T03:54:04.027211+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.4223v3","created_at":"2026-05-18T03:54:04.027211+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.4223","created_at":"2026-05-18T03:54:04.027211+00:00"},{"alias_kind":"pith_short_12","alias_value":"E2ZNQQ4TZKWP","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"E2ZNQQ4TZKWPMKXR","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"E2ZNQQ4T","created_at":"2026-05-18T12:27:04.183437+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV","json":"https://pith.science/pith/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV.json","graph_json":"https://pith.science/api/pith-number/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV/graph.json","events_json":"https://pith.science/api/pith-number/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV/events.json","paper":"https://pith.science/paper/E2ZNQQ4T"},"agent_actions":{"view_html":"https://pith.science/pith/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV","download_json":"https://pith.science/pith/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV.json","view_paper":"https://pith.science/paper/E2ZNQQ4T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.4223&json=true","fetch_graph":"https://pith.science/api/pith-number/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV/graph.json","fetch_events":"https://pith.science/api/pith-number/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV/action/storage_attestation","attest_author":"https://pith.science/pith/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV/action/author_attestation","sign_citation":"https://pith.science/pith/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV/action/citation_signature","submit_replication":"https://pith.science/pith/E2ZNQQ4TZKWPMKXR3H3MWNQ2GV/action/replication_record"}},"created_at":"2026-05-18T03:54:04.027211+00:00","updated_at":"2026-05-18T03:54:04.027211+00:00"}