{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:7DNNAGUSCKGDEJAZX4CX53CMBL","short_pith_number":"pith:7DNNAGUS","canonical_record":{"source":{"id":"1102.0552","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-02T20:48:39Z","cross_cats_sorted":[],"title_canon_sha256":"3b60798d7d532c29ac51eb97763c5d8e9712f39277f1d21bd3fa56a18e2e5f7c","abstract_canon_sha256":"23f41a4d160e1f19c8aa5c5b712d7ec0298b80942e1dd831c46f811ac001cccd"},"schema_version":"1.0"},"canonical_sha256":"f8dad01a92128c322419bf057eec4c0afbf35b5ad42e6ac8a8bcaa3715200d1a","source":{"kind":"arxiv","id":"1102.0552","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.0552","created_at":"2026-05-18T04:30:04Z"},{"alias_kind":"arxiv_version","alias_value":"1102.0552v1","created_at":"2026-05-18T04:30:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.0552","created_at":"2026-05-18T04:30:04Z"},{"alias_kind":"pith_short_12","alias_value":"7DNNAGUSCKGD","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7DNNAGUSCKGDEJAZ","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7DNNAGUS","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:7DNNAGUSCKGDEJAZX4CX53CMBL","target":"record","payload":{"canonical_record":{"source":{"id":"1102.0552","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-02T20:48:39Z","cross_cats_sorted":[],"title_canon_sha256":"3b60798d7d532c29ac51eb97763c5d8e9712f39277f1d21bd3fa56a18e2e5f7c","abstract_canon_sha256":"23f41a4d160e1f19c8aa5c5b712d7ec0298b80942e1dd831c46f811ac001cccd"},"schema_version":"1.0"},"canonical_sha256":"f8dad01a92128c322419bf057eec4c0afbf35b5ad42e6ac8a8bcaa3715200d1a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:30:04.801523Z","signature_b64":"xNN513QtZlxHs7GcMZoJAuTL2Nr2aWzffM+jMYCSH+vXrwq5jy0gjt8nSU7vIMA2qd0eczV3qvRgmkDatpcPDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8dad01a92128c322419bf057eec4c0afbf35b5ad42e6ac8a8bcaa3715200d1a","last_reissued_at":"2026-05-18T04:30:04.801115Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:30:04.801115Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1102.0552","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":"2026-05-18T04:30:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"04oj9cc2kA4gynHXwLPEO57GbwD0DV2BVtArRVGmD1gQtaZfHyEHLURhOCVLX8qZpn7eovn5mSsSO7j+XOrvDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:26:27.059709Z"},"content_sha256":"ac41b46264d9d458f05ec8f7fb4a4c08d6ebbf19af7392d11433af45c205f89f","schema_version":"1.0","event_id":"sha256:ac41b46264d9d458f05ec8f7fb4a4c08d6ebbf19af7392d11433af45c205f89f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:7DNNAGUSCKGDEJAZX4CX53CMBL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Forcing large complete minors in infinite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jos\\'e Zamora, Maya Stein","submitted_at":"2011-02-02T20:48:39Z","abstract_excerpt":"It is well-known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalise the notion of the relative end degree, which had been previously introduced by the first author for locally finite graphs, and show that large minimum relative degree at the ends and large minimum degree at the vertices imply the existence of large complete (topological) minors in infinite graphs with countably many ends."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0552","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":""},"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":"2026-05-18T04:30:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w/pfKK5CTrHoF3XAuafWMkQHF7DHr5mLPqyU53F3eGDvMXUfFm1S04NLCZNoosClQ55GJamTSRODr4D3xtJICw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:26:27.060047Z"},"content_sha256":"6729592993921c917ec15c15b293daabaf5aa29086e50d22431ca6562df769fd","schema_version":"1.0","event_id":"sha256:6729592993921c917ec15c15b293daabaf5aa29086e50d22431ca6562df769fd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7DNNAGUSCKGDEJAZX4CX53CMBL/bundle.json","state_url":"https://pith.science/pith/7DNNAGUSCKGDEJAZX4CX53CMBL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7DNNAGUSCKGDEJAZX4CX53CMBL/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":"2026-06-02T22:26:27Z","links":{"resolver":"https://pith.science/pith/7DNNAGUSCKGDEJAZX4CX53CMBL","bundle":"https://pith.science/pith/7DNNAGUSCKGDEJAZX4CX53CMBL/bundle.json","state":"https://pith.science/pith/7DNNAGUSCKGDEJAZX4CX53CMBL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7DNNAGUSCKGDEJAZX4CX53CMBL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:7DNNAGUSCKGDEJAZX4CX53CMBL","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":"23f41a4d160e1f19c8aa5c5b712d7ec0298b80942e1dd831c46f811ac001cccd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-02T20:48:39Z","title_canon_sha256":"3b60798d7d532c29ac51eb97763c5d8e9712f39277f1d21bd3fa56a18e2e5f7c"},"schema_version":"1.0","source":{"id":"1102.0552","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.0552","created_at":"2026-05-18T04:30:04Z"},{"alias_kind":"arxiv_version","alias_value":"1102.0552v1","created_at":"2026-05-18T04:30:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.0552","created_at":"2026-05-18T04:30:04Z"},{"alias_kind":"pith_short_12","alias_value":"7DNNAGUSCKGD","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7DNNAGUSCKGDEJAZ","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7DNNAGUS","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:6729592993921c917ec15c15b293daabaf5aa29086e50d22431ca6562df769fd","target":"graph","created_at":"2026-05-18T04:30:04Z","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":"It is well-known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalise the notion of the relative end degree, which had been previously introduced by the first author for locally finite graphs, and show that large minimum relative degree at the ends and large minimum degree at the vertices imply the existence of large complete (topological) minors in infinite graphs with countably many ends.","authors_text":"Jos\\'e Zamora, Maya Stein","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-02T20:48:39Z","title":"Forcing large complete minors in infinite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0552","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:ac41b46264d9d458f05ec8f7fb4a4c08d6ebbf19af7392d11433af45c205f89f","target":"record","created_at":"2026-05-18T04:30:04Z","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":"23f41a4d160e1f19c8aa5c5b712d7ec0298b80942e1dd831c46f811ac001cccd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-02T20:48:39Z","title_canon_sha256":"3b60798d7d532c29ac51eb97763c5d8e9712f39277f1d21bd3fa56a18e2e5f7c"},"schema_version":"1.0","source":{"id":"1102.0552","kind":"arxiv","version":1}},"canonical_sha256":"f8dad01a92128c322419bf057eec4c0afbf35b5ad42e6ac8a8bcaa3715200d1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8dad01a92128c322419bf057eec4c0afbf35b5ad42e6ac8a8bcaa3715200d1a","first_computed_at":"2026-05-18T04:30:04.801115Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:30:04.801115Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xNN513QtZlxHs7GcMZoJAuTL2Nr2aWzffM+jMYCSH+vXrwq5jy0gjt8nSU7vIMA2qd0eczV3qvRgmkDatpcPDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:30:04.801523Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.0552","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ac41b46264d9d458f05ec8f7fb4a4c08d6ebbf19af7392d11433af45c205f89f","sha256:6729592993921c917ec15c15b293daabaf5aa29086e50d22431ca6562df769fd"],"state_sha256":"2de17429182c97da2442a07b4093fad311712aae6a4c0fe2a2c6c218751adb90"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6tRyMWygo80pPa64VSOPzIpsK8u4phYSe1o7d07u0Go50qNmUcw03ZUp7L1zB2CwCR0hBvj3exdAz6QG6NilBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T22:26:27.062007Z","bundle_sha256":"0377b50544980fdf6e75e956e2f40e3c38c9814214e28609a1ab076e402c3ea3"}}