{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:JTT44KCH6KW5FKNH6FYPCI6Z7Y","short_pith_number":"pith:JTT44KCH","canonical_record":{"source":{"id":"1101.2532","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-13T10:43:04Z","cross_cats_sorted":[],"title_canon_sha256":"9c6e6b5c10d44879742182fd66892e78f21473513448735a2b79d11393197799","abstract_canon_sha256":"867c72c5b446320e94df02d46e5feaaa462267c83fc3ca512302df7eb92cfdb3"},"schema_version":"1.0"},"canonical_sha256":"4ce7ce2847f2add2a9a7f170f123d9fe31f16973f1c50c3f77548e648ac675f4","source":{"kind":"arxiv","id":"1101.2532","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.2532","created_at":"2026-05-18T03:13:56Z"},{"alias_kind":"arxiv_version","alias_value":"1101.2532v3","created_at":"2026-05-18T03:13:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.2532","created_at":"2026-05-18T03:13:56Z"},{"alias_kind":"pith_short_12","alias_value":"JTT44KCH6KW5","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JTT44KCH6KW5FKNH","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JTT44KCH","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:JTT44KCH6KW5FKNH6FYPCI6Z7Y","target":"record","payload":{"canonical_record":{"source":{"id":"1101.2532","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-13T10:43:04Z","cross_cats_sorted":[],"title_canon_sha256":"9c6e6b5c10d44879742182fd66892e78f21473513448735a2b79d11393197799","abstract_canon_sha256":"867c72c5b446320e94df02d46e5feaaa462267c83fc3ca512302df7eb92cfdb3"},"schema_version":"1.0"},"canonical_sha256":"4ce7ce2847f2add2a9a7f170f123d9fe31f16973f1c50c3f77548e648ac675f4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:56.553321Z","signature_b64":"AbYDCo5LR4YRKEn6cPAOA9JaW39t4mvhOxCT6n6RXlAX5H8pFUuVFQQRRRHDM2RmYAW9FMpCYE+U06n4J5e5AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ce7ce2847f2add2a9a7f170f123d9fe31f16973f1c50c3f77548e648ac675f4","last_reissued_at":"2026-05-18T03:13:56.552680Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:56.552680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1101.2532","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:13:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nv9rkA+eWZYZcwiBDvyYMhIBMCRubF2bRg//JfwUdm3IkCpbm8jmuZW8x1vSAiZ1N/gqlLirQ7JguLqdm6h1Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T01:25:53.413362Z"},"content_sha256":"545eed26c8b5294e11425474f62f636809a6381e0d59f317e40c7ddfc7bafb29","schema_version":"1.0","event_id":"sha256:545eed26c8b5294e11425474f62f636809a6381e0d59f317e40c7ddfc7bafb29"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:JTT44KCH6KW5FKNH6FYPCI6Z7Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Pl\\\"unnecke's Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Giorgis Petridis","submitted_at":"2011-01-13T10:43:04Z","abstract_excerpt":"Plunnecke's inequality is the standard tool to obtain estimates on the cardinality of sumsets and has many applications in additive combinatorics. We present a new proof. The main novelty is that the proof is completed with no reference to Menger's theorem or Cartesian products of graphs. We also investigate the sharpness of the inequality and show that it can be sharp for arbitrarily long, but not for infinite commutative graphs. A key step in our investigation is the construction of arbitrarily long regular commutative graphs. Lastly we prove a necessary condition for the inequality to be at"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.2532","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:13:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T8cdPB9Z1guBUDvvC7vrOK30qcraCkN6kmz6WDAhs2+8xD9y6bU3vwRcNBG495OZJI7fA0qw5OwCkmj3Or4tAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T01:25:53.413742Z"},"content_sha256":"e03d8fed44040cff67e28e8f52fe61c97b79bb0bb62602807e1d37bcf008594c","schema_version":"1.0","event_id":"sha256:e03d8fed44040cff67e28e8f52fe61c97b79bb0bb62602807e1d37bcf008594c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JTT44KCH6KW5FKNH6FYPCI6Z7Y/bundle.json","state_url":"https://pith.science/pith/JTT44KCH6KW5FKNH6FYPCI6Z7Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JTT44KCH6KW5FKNH6FYPCI6Z7Y/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":"2026-06-26T01:25:53Z","links":{"resolver":"https://pith.science/pith/JTT44KCH6KW5FKNH6FYPCI6Z7Y","bundle":"https://pith.science/pith/JTT44KCH6KW5FKNH6FYPCI6Z7Y/bundle.json","state":"https://pith.science/pith/JTT44KCH6KW5FKNH6FYPCI6Z7Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JTT44KCH6KW5FKNH6FYPCI6Z7Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:JTT44KCH6KW5FKNH6FYPCI6Z7Y","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":"867c72c5b446320e94df02d46e5feaaa462267c83fc3ca512302df7eb92cfdb3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-13T10:43:04Z","title_canon_sha256":"9c6e6b5c10d44879742182fd66892e78f21473513448735a2b79d11393197799"},"schema_version":"1.0","source":{"id":"1101.2532","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.2532","created_at":"2026-05-18T03:13:56Z"},{"alias_kind":"arxiv_version","alias_value":"1101.2532v3","created_at":"2026-05-18T03:13:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.2532","created_at":"2026-05-18T03:13:56Z"},{"alias_kind":"pith_short_12","alias_value":"JTT44KCH6KW5","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JTT44KCH6KW5FKNH","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JTT44KCH","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:e03d8fed44040cff67e28e8f52fe61c97b79bb0bb62602807e1d37bcf008594c","target":"graph","created_at":"2026-05-18T03:13:56Z","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":"Plunnecke's inequality is the standard tool to obtain estimates on the cardinality of sumsets and has many applications in additive combinatorics. We present a new proof. The main novelty is that the proof is completed with no reference to Menger's theorem or Cartesian products of graphs. We also investigate the sharpness of the inequality and show that it can be sharp for arbitrarily long, but not for infinite commutative graphs. A key step in our investigation is the construction of arbitrarily long regular commutative graphs. Lastly we prove a necessary condition for the inequality to be at","authors_text":"Giorgis Petridis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-13T10:43:04Z","title":"Pl\\\"unnecke's Inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.2532","kind":"arxiv","version":3},"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":"sha256:545eed26c8b5294e11425474f62f636809a6381e0d59f317e40c7ddfc7bafb29","target":"record","created_at":"2026-05-18T03:13:56Z","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":"867c72c5b446320e94df02d46e5feaaa462267c83fc3ca512302df7eb92cfdb3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-13T10:43:04Z","title_canon_sha256":"9c6e6b5c10d44879742182fd66892e78f21473513448735a2b79d11393197799"},"schema_version":"1.0","source":{"id":"1101.2532","kind":"arxiv","version":3}},"canonical_sha256":"4ce7ce2847f2add2a9a7f170f123d9fe31f16973f1c50c3f77548e648ac675f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4ce7ce2847f2add2a9a7f170f123d9fe31f16973f1c50c3f77548e648ac675f4","first_computed_at":"2026-05-18T03:13:56.552680Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:56.552680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AbYDCo5LR4YRKEn6cPAOA9JaW39t4mvhOxCT6n6RXlAX5H8pFUuVFQQRRRHDM2RmYAW9FMpCYE+U06n4J5e5AA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:56.553321Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.2532","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:545eed26c8b5294e11425474f62f636809a6381e0d59f317e40c7ddfc7bafb29","sha256:e03d8fed44040cff67e28e8f52fe61c97b79bb0bb62602807e1d37bcf008594c"],"state_sha256":"d4828e58516a1a353a0a61bb583dc81c266eb4d482312b7b702c451dc0633fe9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"59Cq/qYZ2ONmpQ6DbJMHsYzoiVcbKVk6K6aAePj7qX7Kr8qvAuemYZ0fgGfMnp48Hk2qM43YXIi+iF5QnteGDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T01:25:53.415654Z","bundle_sha256":"c20bab195ce0d25910be4925f9e2caa41cf27ad43a5479caa625b1af05b18402"}}