{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:XKL4HPQZRZPSI7IJIPJETIX5UA","short_pith_number":"pith:XKL4HPQZ","canonical_record":{"source":{"id":"math/0702354","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-02-13T04:18:16Z","cross_cats_sorted":[],"title_canon_sha256":"af0df3319c66b4937d87cd3b2bdb32e372c7c0af44387923fd3012a8332bf6c4","abstract_canon_sha256":"5712f17c03f330843812b04fc97e337039f42fd5eee494e9d413d0c535fbbad0"},"schema_version":"1.0"},"canonical_sha256":"ba97c3be198e5f247d0943d249a2fda03c6e446b85f40f6022fd152fa397ed38","source":{"kind":"arxiv","id":"math/0702354","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0702354","created_at":"2026-07-04T14:59:31Z"},{"alias_kind":"arxiv_version","alias_value":"math/0702354v1","created_at":"2026-07-04T14:59:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0702354","created_at":"2026-07-04T14:59:31Z"},{"alias_kind":"pith_short_12","alias_value":"XKL4HPQZRZPS","created_at":"2026-07-04T14:59:31Z"},{"alias_kind":"pith_short_16","alias_value":"XKL4HPQZRZPSI7IJ","created_at":"2026-07-04T14:59:31Z"},{"alias_kind":"pith_short_8","alias_value":"XKL4HPQZ","created_at":"2026-07-04T14:59:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:XKL4HPQZRZPSI7IJIPJETIX5UA","target":"record","payload":{"canonical_record":{"source":{"id":"math/0702354","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-02-13T04:18:16Z","cross_cats_sorted":[],"title_canon_sha256":"af0df3319c66b4937d87cd3b2bdb32e372c7c0af44387923fd3012a8332bf6c4","abstract_canon_sha256":"5712f17c03f330843812b04fc97e337039f42fd5eee494e9d413d0c535fbbad0"},"schema_version":"1.0"},"canonical_sha256":"ba97c3be198e5f247d0943d249a2fda03c6e446b85f40f6022fd152fa397ed38","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:59:31.105303Z","signature_b64":"cl/0f+CHjxVz2abgEUzMbY5OzAzQUR+Xgxb44qMe1zp7nocOiZRadYNPYHy73IWwNsGCjI9kII5Yrkt5fM5UBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba97c3be198e5f247d0943d249a2fda03c6e446b85f40f6022fd152fa397ed38","last_reissued_at":"2026-07-04T14:59:31.104971Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:59:31.104971Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0702354","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":"2026-07-04T14:59:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GPTNQrbQoMvtY+Xfmg+D3mh7RyGsCl33Yo2O+LpXfyOF/9ZNq+SyQgCY8mGwhI+PiM8rRgRFLTroAbflxzXRCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T05:39:57.609329Z"},"content_sha256":"cb9a66b66a25f55d54a2f45fe392b483af932b0a6c5b1970739810c9e8eb986c","schema_version":"1.0","event_id":"sha256:cb9a66b66a25f55d54a2f45fe392b483af932b0a6c5b1970739810c9e8eb986c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:XKL4HPQZRZPSI7IJIPJETIX5UA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Highly connected monochromatic subgraphs of multicoloured graphs","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Henry Liu, Noah Prince, Robert Morris","submitted_at":"2007-02-13T04:18:16Z","abstract_excerpt":"We consider the following question of Bollobas: given an r-colouring of the edges of the complete graph on n vertices, how large a k-connected subgraph can we find using only one colour? We solve this problem asymptotically when r-1 is a prime power, and exactly for 2- and 3-colourings."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702354","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/math/0702354/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":"2026-07-04T14:59:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0WsLSkLo7to95tCzj9MGp3sSwSYzjThWRuKYHn8sz4n6Vgi+6pwshZb3TyrX3ZkjNTqxm1xI0VBLVqUc4ak4Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T05:39:57.609722Z"},"content_sha256":"7e6654c97b6070de35801136702d10fb516443c3f4f7a3556a317a202707dc59","schema_version":"1.0","event_id":"sha256:7e6654c97b6070de35801136702d10fb516443c3f4f7a3556a317a202707dc59"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XKL4HPQZRZPSI7IJIPJETIX5UA/bundle.json","state_url":"https://pith.science/pith/XKL4HPQZRZPSI7IJIPJETIX5UA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XKL4HPQZRZPSI7IJIPJETIX5UA/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":"2026-07-05T05:39:57Z","links":{"resolver":"https://pith.science/pith/XKL4HPQZRZPSI7IJIPJETIX5UA","bundle":"https://pith.science/pith/XKL4HPQZRZPSI7IJIPJETIX5UA/bundle.json","state":"https://pith.science/pith/XKL4HPQZRZPSI7IJIPJETIX5UA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XKL4HPQZRZPSI7IJIPJETIX5UA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:XKL4HPQZRZPSI7IJIPJETIX5UA","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":"5712f17c03f330843812b04fc97e337039f42fd5eee494e9d413d0c535fbbad0","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2007-02-13T04:18:16Z","title_canon_sha256":"af0df3319c66b4937d87cd3b2bdb32e372c7c0af44387923fd3012a8332bf6c4"},"schema_version":"1.0","source":{"id":"math/0702354","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0702354","created_at":"2026-07-04T14:59:31Z"},{"alias_kind":"arxiv_version","alias_value":"math/0702354v1","created_at":"2026-07-04T14:59:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0702354","created_at":"2026-07-04T14:59:31Z"},{"alias_kind":"pith_short_12","alias_value":"XKL4HPQZRZPS","created_at":"2026-07-04T14:59:31Z"},{"alias_kind":"pith_short_16","alias_value":"XKL4HPQZRZPSI7IJ","created_at":"2026-07-04T14:59:31Z"},{"alias_kind":"pith_short_8","alias_value":"XKL4HPQZ","created_at":"2026-07-04T14:59:31Z"}],"graph_snapshots":[{"event_id":"sha256:7e6654c97b6070de35801136702d10fb516443c3f4f7a3556a317a202707dc59","target":"graph","created_at":"2026-07-04T14:59:31Z","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/math/0702354/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider the following question of Bollobas: given an r-colouring of the edges of the complete graph on n vertices, how large a k-connected subgraph can we find using only one colour? We solve this problem asymptotically when r-1 is a prime power, and exactly for 2- and 3-colourings.","authors_text":"Henry Liu, Noah Prince, Robert Morris","cross_cats":[],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2007-02-13T04:18:16Z","title":"Highly connected monochromatic subgraphs of multicoloured graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702354","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:cb9a66b66a25f55d54a2f45fe392b483af932b0a6c5b1970739810c9e8eb986c","target":"record","created_at":"2026-07-04T14:59:31Z","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":"5712f17c03f330843812b04fc97e337039f42fd5eee494e9d413d0c535fbbad0","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2007-02-13T04:18:16Z","title_canon_sha256":"af0df3319c66b4937d87cd3b2bdb32e372c7c0af44387923fd3012a8332bf6c4"},"schema_version":"1.0","source":{"id":"math/0702354","kind":"arxiv","version":1}},"canonical_sha256":"ba97c3be198e5f247d0943d249a2fda03c6e446b85f40f6022fd152fa397ed38","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ba97c3be198e5f247d0943d249a2fda03c6e446b85f40f6022fd152fa397ed38","first_computed_at":"2026-07-04T14:59:31.104971Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:59:31.104971Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cl/0f+CHjxVz2abgEUzMbY5OzAzQUR+Xgxb44qMe1zp7nocOiZRadYNPYHy73IWwNsGCjI9kII5Yrkt5fM5UBw==","signature_status":"signed_v1","signed_at":"2026-07-04T14:59:31.105303Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0702354","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb9a66b66a25f55d54a2f45fe392b483af932b0a6c5b1970739810c9e8eb986c","sha256:7e6654c97b6070de35801136702d10fb516443c3f4f7a3556a317a202707dc59"],"state_sha256":"f7f881bd14af7f205b627538d22279d5ddeeba75dc5b4d31876f9cefcca17573"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fUomjUsFBZC7h8wGzfdqn0l94pzuxa0A/nRmGHI0gIPjQ5BdvGYSeQkKZLPx/5urDZ97xlBxj7njX7p02oszDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T05:39:57.611655Z","bundle_sha256":"09cd91ec12c38c88487bf0e6ef7078a8225684fcb6216cc05bede8f929fab3fb"}}