{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:AFUWHATI5CPYSUMFTZJR52JQOX","short_pith_number":"pith:AFUWHATI","canonical_record":{"source":{"id":"2007.04478","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-07-08T23:50:43Z","cross_cats_sorted":[],"title_canon_sha256":"b8a4ea2caf01ba8670cb2f3980eefd4a95247e3e4c89ff6e10aa82b575d3b019","abstract_canon_sha256":"c5d9612989fdd8e6a7d36affe80bc5b69927e540ee6faf75910c117f007c7754"},"schema_version":"1.0"},"canonical_sha256":"0169638268e89f8951859e531ee93075d269adfaa840957d6f8a2f2acc6e7a12","source":{"kind":"arxiv","id":"2007.04478","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.04478","created_at":"2026-07-05T01:17:30Z"},{"alias_kind":"arxiv_version","alias_value":"2007.04478v1","created_at":"2026-07-05T01:17:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.04478","created_at":"2026-07-05T01:17:30Z"},{"alias_kind":"pith_short_12","alias_value":"AFUWHATI5CPY","created_at":"2026-07-05T01:17:30Z"},{"alias_kind":"pith_short_16","alias_value":"AFUWHATI5CPYSUMF","created_at":"2026-07-05T01:17:30Z"},{"alias_kind":"pith_short_8","alias_value":"AFUWHATI","created_at":"2026-07-05T01:17:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:AFUWHATI5CPYSUMFTZJR52JQOX","target":"record","payload":{"canonical_record":{"source":{"id":"2007.04478","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-07-08T23:50:43Z","cross_cats_sorted":[],"title_canon_sha256":"b8a4ea2caf01ba8670cb2f3980eefd4a95247e3e4c89ff6e10aa82b575d3b019","abstract_canon_sha256":"c5d9612989fdd8e6a7d36affe80bc5b69927e540ee6faf75910c117f007c7754"},"schema_version":"1.0"},"canonical_sha256":"0169638268e89f8951859e531ee93075d269adfaa840957d6f8a2f2acc6e7a12","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:17:30.489094Z","signature_b64":"f/cSGbJOza0XJ2g6u+hjplaOYMOFCR4uxTTXg0XgCGYJFfPf7cXJ7tVvz1uIftMN7xnzCmDYTrItAbkF6hMLCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0169638268e89f8951859e531ee93075d269adfaa840957d6f8a2f2acc6e7a12","last_reissued_at":"2026-07-05T01:17:30.488662Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:17:30.488662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2007.04478","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:17:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xV42QuO7gpxrcOINzf3htoRxPgGXKrAxfTEBOPJLFmIs857Wuab/kHPBl1L6S2BhFwllYjpothwD31SOwoSpCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T04:57:33.430970Z"},"content_sha256":"30d1d071312ea382ebf3dd6d0beb45dc6b34981ac070beda28eddf6c7b6560d4","schema_version":"1.0","event_id":"sha256:30d1d071312ea382ebf3dd6d0beb45dc6b34981ac070beda28eddf6c7b6560d4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:AFUWHATI5CPYSUMFTZJR52JQOX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Closing the Random Graph Gap in Tuza's Conjecture Through the Online Triangle Packing Process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrzej Dudek, Patrick Bennett, Ryan Cushman","submitted_at":"2020-07-08T23:50:43Z","abstract_excerpt":"A long-standing conjecture of Zsolt Tuza asserts that the triangle covering number $\\tau(G)$ is at most twice the triangle packing number $\\nu(G)$, where the triangle packing number $\\nu(G)$ is the maximum size of a set of edge-disjoint triangles in $G$ and the triangle covering number $\\tau(G)$ is the minimal size of a set of edges intersecting all triangles. In this paper, we prove that Tuza's conjecture holds in the Erd\\H{o}s-R\\'enyi random graph $G(n,m)$ for all range of $m$, closing the gap in what was previously known. (Recently, this result was also independently proved by Jeff Kahn and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.04478","kind":"arxiv","version":1},"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/2007.04478/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":"2026-07-05T01:17:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sS3KERZQcgsqb9WMJ04Hz5PO2iQR91F3l49QFUCK9crIp68dcBvZLgvtQL6ZioGXvnZteNjQB26wQc3rsiPbBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T04:57:33.431735Z"},"content_sha256":"76285a734547742c9d00c27377ef74b0841a71d725609e0419e653925e3f64e5","schema_version":"1.0","event_id":"sha256:76285a734547742c9d00c27377ef74b0841a71d725609e0419e653925e3f64e5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AFUWHATI5CPYSUMFTZJR52JQOX/bundle.json","state_url":"https://pith.science/pith/AFUWHATI5CPYSUMFTZJR52JQOX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AFUWHATI5CPYSUMFTZJR52JQOX/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":"2026-08-19T04:57:33Z","links":{"resolver":"https://pith.science/pith/AFUWHATI5CPYSUMFTZJR52JQOX","bundle":"https://pith.science/pith/AFUWHATI5CPYSUMFTZJR52JQOX/bundle.json","state":"https://pith.science/pith/AFUWHATI5CPYSUMFTZJR52JQOX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AFUWHATI5CPYSUMFTZJR52JQOX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:AFUWHATI5CPYSUMFTZJR52JQOX","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":"c5d9612989fdd8e6a7d36affe80bc5b69927e540ee6faf75910c117f007c7754","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-07-08T23:50:43Z","title_canon_sha256":"b8a4ea2caf01ba8670cb2f3980eefd4a95247e3e4c89ff6e10aa82b575d3b019"},"schema_version":"1.0","source":{"id":"2007.04478","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.04478","created_at":"2026-07-05T01:17:30Z"},{"alias_kind":"arxiv_version","alias_value":"2007.04478v1","created_at":"2026-07-05T01:17:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.04478","created_at":"2026-07-05T01:17:30Z"},{"alias_kind":"pith_short_12","alias_value":"AFUWHATI5CPY","created_at":"2026-07-05T01:17:30Z"},{"alias_kind":"pith_short_16","alias_value":"AFUWHATI5CPYSUMF","created_at":"2026-07-05T01:17:30Z"},{"alias_kind":"pith_short_8","alias_value":"AFUWHATI","created_at":"2026-07-05T01:17:30Z"}],"graph_snapshots":[{"event_id":"sha256:76285a734547742c9d00c27377ef74b0841a71d725609e0419e653925e3f64e5","target":"graph","created_at":"2026-07-05T01:17:30Z","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/2007.04478/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A long-standing conjecture of Zsolt Tuza asserts that the triangle covering number $\\tau(G)$ is at most twice the triangle packing number $\\nu(G)$, where the triangle packing number $\\nu(G)$ is the maximum size of a set of edge-disjoint triangles in $G$ and the triangle covering number $\\tau(G)$ is the minimal size of a set of edges intersecting all triangles. In this paper, we prove that Tuza's conjecture holds in the Erd\\H{o}s-R\\'enyi random graph $G(n,m)$ for all range of $m$, closing the gap in what was previously known. (Recently, this result was also independently proved by Jeff Kahn and","authors_text":"Andrzej Dudek, Patrick Bennett, Ryan Cushman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-07-08T23:50:43Z","title":"Closing the Random Graph Gap in Tuza's Conjecture Through the Online Triangle Packing Process"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.04478","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":"sha256:30d1d071312ea382ebf3dd6d0beb45dc6b34981ac070beda28eddf6c7b6560d4","target":"record","created_at":"2026-07-05T01:17:30Z","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":"c5d9612989fdd8e6a7d36affe80bc5b69927e540ee6faf75910c117f007c7754","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-07-08T23:50:43Z","title_canon_sha256":"b8a4ea2caf01ba8670cb2f3980eefd4a95247e3e4c89ff6e10aa82b575d3b019"},"schema_version":"1.0","source":{"id":"2007.04478","kind":"arxiv","version":1}},"canonical_sha256":"0169638268e89f8951859e531ee93075d269adfaa840957d6f8a2f2acc6e7a12","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0169638268e89f8951859e531ee93075d269adfaa840957d6f8a2f2acc6e7a12","first_computed_at":"2026-07-05T01:17:30.488662Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:17:30.488662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"f/cSGbJOza0XJ2g6u+hjplaOYMOFCR4uxTTXg0XgCGYJFfPf7cXJ7tVvz1uIftMN7xnzCmDYTrItAbkF6hMLCw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:17:30.489094Z","signed_message":"canonical_sha256_bytes"},"source_id":"2007.04478","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:30d1d071312ea382ebf3dd6d0beb45dc6b34981ac070beda28eddf6c7b6560d4","sha256:76285a734547742c9d00c27377ef74b0841a71d725609e0419e653925e3f64e5"],"state_sha256":"f9a23a9627987d4904e4a11da772644304cb1b503ab99bef4e466e8469e95da8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7ztGBo+ngfmdHpBMN2C23uAqLELNm3w7c71thkLmtXS8ZdbZehUUsXbFYKbUfaVZZvma1E3KYBozoqrvKZfeAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T04:57:33.437818Z","bundle_sha256":"b510282d87a84071004f3e4b87f4ce37fcd85c0afce20f9800749c226190596b"}}