{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2DT6XT2Y2IJVQY5UQBZSRAHFZS","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":"9d422a8c5f29d2e4c8aedf5dc5db351249d4b301112eeadd5078a38e99d1c336","cross_cats_sorted":["cs.DS","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-07-10T15:04:33Z","title_canon_sha256":"faf1b7460217c878d8faa08ef838cc218e8e2dadda70f1f37e2028643154435e"},"schema_version":"1.0","source":{"id":"1507.02929","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.02929","created_at":"2026-05-18T01:37:02Z"},{"alias_kind":"arxiv_version","alias_value":"1507.02929v1","created_at":"2026-05-18T01:37:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.02929","created_at":"2026-05-18T01:37:02Z"},{"alias_kind":"pith_short_12","alias_value":"2DT6XT2Y2IJV","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"2DT6XT2Y2IJVQY5U","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"2DT6XT2Y","created_at":"2026-05-18T12:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:f0e1b4b5f7c0c91d6b5f0bd55e2a5d13ad7cce4193ab66448256d851453d2576","target":"graph","created_at":"2026-05-18T01:37:02Z","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":"Planar Maximally Filtered Graphs (PMFG) are an important tool for filtering the most relevant information from correlation based networks such as stock market networks. One of the main characteristics of a PMFG is the number of its 3- and 4-cliques. Recently in a few high impact papers it was stated that, based on heuristic evidence, the maximum number of 3- and 4-cliques that can exist in a PMFG with n vertices is 3n - 8 and n - 4 respectively. In this paper, we prove that this is indeed the case.","authors_text":"Athanasios A. Pantelous, Jenna Birch, Konstantin Zuev","cross_cats":["cs.DS","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-07-10T15:04:33Z","title":"The Maximum Number of 3- and 4-Cliques within a Planar Maximally Filtered Graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02929","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":"sha256:4f0bd39e2a56bfe9ae9cdf217aff017f767fd7989877d1684a6b2f6508ae4218","target":"record","created_at":"2026-05-18T01:37:02Z","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":"9d422a8c5f29d2e4c8aedf5dc5db351249d4b301112eeadd5078a38e99d1c336","cross_cats_sorted":["cs.DS","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-07-10T15:04:33Z","title_canon_sha256":"faf1b7460217c878d8faa08ef838cc218e8e2dadda70f1f37e2028643154435e"},"schema_version":"1.0","source":{"id":"1507.02929","kind":"arxiv","version":1}},"canonical_sha256":"d0e7ebcf58d2135863b480732880e5cc934ee091a3d33000e6c7d9b2cda4a5cd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d0e7ebcf58d2135863b480732880e5cc934ee091a3d33000e6c7d9b2cda4a5cd","first_computed_at":"2026-05-18T01:37:02.912569Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:02.912569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JReuQ+VIJeXnnXCGnmnBnR817c3WCIwgSxFue6C4P5CMCWkQkYfnrVHH7tYIhQs8EDMi6ztHV2YJk9sENgZ8CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:02.912996Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.02929","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f0bd39e2a56bfe9ae9cdf217aff017f767fd7989877d1684a6b2f6508ae4218","sha256:f0e1b4b5f7c0c91d6b5f0bd55e2a5d13ad7cce4193ab66448256d851453d2576"],"state_sha256":"03974d3e50edf4da107b1e9c7356174d94455b95e1e107cabf667cdf9c2deb3e"}