{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:WRYJNVFTNBN7ZZ37K3F6WVH3MK","short_pith_number":"pith:WRYJNVFT","canonical_record":{"source":{"id":"1407.7707","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-29T12:28:50Z","cross_cats_sorted":[],"title_canon_sha256":"4c1bad597ce22fdecbf6dc0809225d1a9ff0e89f8151f5461d14643b2dfe2335","abstract_canon_sha256":"39fc9b81c60378fbf03513193c0bab5d0eb0d50505a1cd5e80a8f37db7a1e589"},"schema_version":"1.0"},"canonical_sha256":"b47096d4b3685bfce77f56cbeb54fb629b9736ceb4a4db0bf95b270532f94a66","source":{"kind":"arxiv","id":"1407.7707","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.7707","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"arxiv_version","alias_value":"1407.7707v5","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.7707","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"pith_short_12","alias_value":"WRYJNVFTNBN7","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"WRYJNVFTNBN7ZZ37","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"WRYJNVFT","created_at":"2026-05-18T12:28:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:WRYJNVFTNBN7ZZ37K3F6WVH3MK","target":"record","payload":{"canonical_record":{"source":{"id":"1407.7707","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-29T12:28:50Z","cross_cats_sorted":[],"title_canon_sha256":"4c1bad597ce22fdecbf6dc0809225d1a9ff0e89f8151f5461d14643b2dfe2335","abstract_canon_sha256":"39fc9b81c60378fbf03513193c0bab5d0eb0d50505a1cd5e80a8f37db7a1e589"},"schema_version":"1.0"},"canonical_sha256":"b47096d4b3685bfce77f56cbeb54fb629b9736ceb4a4db0bf95b270532f94a66","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:03.413299Z","signature_b64":"zlRSDk+TJ/M0IEHDaYLdZePm2mXLIJLFQHv+9ifdDchVXYj2aUs2X0QdbdBfoj5J11Q8RyeqtJcxMfWKzNImDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b47096d4b3685bfce77f56cbeb54fb629b9736ceb4a4db0bf95b270532f94a66","last_reissued_at":"2026-05-18T00:16:03.412877Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:03.412877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.7707","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:16:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I/V7BqC0z8f5XM4nENknF/xGKrXMtNsl6O0uGyDv8JZSJBR4FtO9usJNHszfvSp74vOLDl2tiC48HJnIchf9Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T04:42:01.505450Z"},"content_sha256":"ea8c62a8b8201d1bbb0f5e0b7ea948079af1d0ad01c6b789860e56c7a730b63a","schema_version":"1.0","event_id":"sha256:ea8c62a8b8201d1bbb0f5e0b7ea948079af1d0ad01c6b789860e56c7a730b63a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:WRYJNVFTNBN7ZZ37K3F6WVH3MK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Number of cliques in graphs with a forbidden subdivision","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Choongbum Lee, Sang-il Oum","submitted_at":"2014-07-29T12:28:50Z","abstract_excerpt":"We prove that for all positive integers $t$, every $n$-vertex graph with no $K_t$-subdivision has at most $2^{50t}n$ cliques. We also prove that asymptotically, such graphs contain at most $2^{(5+o(1))t}n$ cliques, where $o(1)$ tends to zero as $t$ tends to infinity. This strongly answers a question of D. Wood asking if the number of cliques in $n$-vertex graphs with no $K_t$-minor is at most $2^{ct}n$ for some constant $c$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7707","kind":"arxiv","version":5},"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":"2026-05-18T00:16:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x4sQQvqH7hQjG0p7oy8T9gaSkn7gYagyVV8dG1WIU7Sn4mCmuKV2dtQd+VoBfKzoXHBwTmwVc5ADW65w/LkSCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T04:42:01.505793Z"},"content_sha256":"be8083335832868f6adcf285f90b2619b46ad7e4443e8f2d0dccb7df4e6a6910","schema_version":"1.0","event_id":"sha256:be8083335832868f6adcf285f90b2619b46ad7e4443e8f2d0dccb7df4e6a6910"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WRYJNVFTNBN7ZZ37K3F6WVH3MK/bundle.json","state_url":"https://pith.science/pith/WRYJNVFTNBN7ZZ37K3F6WVH3MK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WRYJNVFTNBN7ZZ37K3F6WVH3MK/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":"2026-06-12T04:42:01Z","links":{"resolver":"https://pith.science/pith/WRYJNVFTNBN7ZZ37K3F6WVH3MK","bundle":"https://pith.science/pith/WRYJNVFTNBN7ZZ37K3F6WVH3MK/bundle.json","state":"https://pith.science/pith/WRYJNVFTNBN7ZZ37K3F6WVH3MK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WRYJNVFTNBN7ZZ37K3F6WVH3MK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:WRYJNVFTNBN7ZZ37K3F6WVH3MK","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":"39fc9b81c60378fbf03513193c0bab5d0eb0d50505a1cd5e80a8f37db7a1e589","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-29T12:28:50Z","title_canon_sha256":"4c1bad597ce22fdecbf6dc0809225d1a9ff0e89f8151f5461d14643b2dfe2335"},"schema_version":"1.0","source":{"id":"1407.7707","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.7707","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"arxiv_version","alias_value":"1407.7707v5","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.7707","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"pith_short_12","alias_value":"WRYJNVFTNBN7","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"WRYJNVFTNBN7ZZ37","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"WRYJNVFT","created_at":"2026-05-18T12:28:54Z"}],"graph_snapshots":[{"event_id":"sha256:be8083335832868f6adcf285f90b2619b46ad7e4443e8f2d0dccb7df4e6a6910","target":"graph","created_at":"2026-05-18T00:16:03Z","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":"We prove that for all positive integers $t$, every $n$-vertex graph with no $K_t$-subdivision has at most $2^{50t}n$ cliques. We also prove that asymptotically, such graphs contain at most $2^{(5+o(1))t}n$ cliques, where $o(1)$ tends to zero as $t$ tends to infinity. This strongly answers a question of D. Wood asking if the number of cliques in $n$-vertex graphs with no $K_t$-minor is at most $2^{ct}n$ for some constant $c$.","authors_text":"Choongbum Lee, Sang-il Oum","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-29T12:28:50Z","title":"Number of cliques in graphs with a forbidden subdivision"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7707","kind":"arxiv","version":5},"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:ea8c62a8b8201d1bbb0f5e0b7ea948079af1d0ad01c6b789860e56c7a730b63a","target":"record","created_at":"2026-05-18T00:16:03Z","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":"39fc9b81c60378fbf03513193c0bab5d0eb0d50505a1cd5e80a8f37db7a1e589","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-29T12:28:50Z","title_canon_sha256":"4c1bad597ce22fdecbf6dc0809225d1a9ff0e89f8151f5461d14643b2dfe2335"},"schema_version":"1.0","source":{"id":"1407.7707","kind":"arxiv","version":5}},"canonical_sha256":"b47096d4b3685bfce77f56cbeb54fb629b9736ceb4a4db0bf95b270532f94a66","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b47096d4b3685bfce77f56cbeb54fb629b9736ceb4a4db0bf95b270532f94a66","first_computed_at":"2026-05-18T00:16:03.412877Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:03.412877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zlRSDk+TJ/M0IEHDaYLdZePm2mXLIJLFQHv+9ifdDchVXYj2aUs2X0QdbdBfoj5J11Q8RyeqtJcxMfWKzNImDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:03.413299Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.7707","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea8c62a8b8201d1bbb0f5e0b7ea948079af1d0ad01c6b789860e56c7a730b63a","sha256:be8083335832868f6adcf285f90b2619b46ad7e4443e8f2d0dccb7df4e6a6910"],"state_sha256":"c865f6f08ed5fda189ac64554ad28075b77946d8058b3bcc7b36c1eac2178a53"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VotHaPu25asqfimu10Sl07SBNU/MWuvgNPNw+JCKl2yAzKp2/gzbXKH16oXVwKwpS8aO9dnx1KIe40L8cuF/DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T04:42:01.508379Z","bundle_sha256":"3407a259129c9ec0e10124afc4bc1b8b88b5e37020721cb7fe47e32959f297fd"}}