{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:CLSZR5EUPYGAUM4E5GFYJEQ32Z","short_pith_number":"pith:CLSZR5EU","canonical_record":{"source":{"id":"1604.00733","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-04T03:54:06Z","cross_cats_sorted":[],"title_canon_sha256":"37b9ed47c7878c49cefa9e6e4e4b92d820ef4ea7199cf43065863904ac038776","abstract_canon_sha256":"2d8613c4a8f73f2d73ac4bc69a2c81e97de0a255ea447a65c53bc9a5918d99b0"},"schema_version":"1.0"},"canonical_sha256":"12e598f4947e0c0a3384e98b84921bd676fa247d66f537439cf94bfc9f0432bb","source":{"kind":"arxiv","id":"1604.00733","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.00733","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"arxiv_version","alias_value":"1604.00733v4","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.00733","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"pith_short_12","alias_value":"CLSZR5EUPYGA","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CLSZR5EUPYGAUM4E","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CLSZR5EU","created_at":"2026-05-18T12:30:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:CLSZR5EUPYGAUM4E5GFYJEQ32Z","target":"record","payload":{"canonical_record":{"source":{"id":"1604.00733","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-04T03:54:06Z","cross_cats_sorted":[],"title_canon_sha256":"37b9ed47c7878c49cefa9e6e4e4b92d820ef4ea7199cf43065863904ac038776","abstract_canon_sha256":"2d8613c4a8f73f2d73ac4bc69a2c81e97de0a255ea447a65c53bc9a5918d99b0"},"schema_version":"1.0"},"canonical_sha256":"12e598f4947e0c0a3384e98b84921bd676fa247d66f537439cf94bfc9f0432bb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:53.303434Z","signature_b64":"Vair9VBxxPWYh65SqcqKrbduf7CKnLiaduNfL31E0/85hsiumIQFlz4SRYnfjO/GVU6tkZry1Sj+xoiCi5VjDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"12e598f4947e0c0a3384e98b84921bd676fa247d66f537439cf94bfc9f0432bb","last_reissued_at":"2026-05-17T23:48:53.302919Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:53.302919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.00733","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:48:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZCZjashICtph16k/3TCcKbwEKCkmc4JSjJIO5gEk6XoLgsuQgVzqXYtG/Dd175WmlpCTIkrNQEmr6BeM1KHkDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T14:53:19.121822Z"},"content_sha256":"66b96bf69b5d57fa898f34366c18af6383a9497b57fc8954861b1179707ac790","schema_version":"1.0","event_id":"sha256:66b96bf69b5d57fa898f34366c18af6383a9497b57fc8954861b1179707ac790"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:CLSZR5EUPYGAUM4E5GFYJEQ32Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On regularity lemmas and their algorithmic applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jacob Fox, L\\'aszl\\'o Mikl\\'os Lov\\'asz, Yufei Zhao","submitted_at":"2016-04-04T03:54:06Z","abstract_excerpt":"Szemer\\'edi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze--Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.00733","kind":"arxiv","version":4},"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":"2026-05-17T23:48:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BQoppXcOPABBlBbQNiGd1JK3qegqdRnz91C8LejYYw5nhliFEGobb6D9Cz+ryLJLwDvhwnss+Dv1QSPDwVbsBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T14:53:19.122535Z"},"content_sha256":"d56b0e2198c2aa40b440a285311c04c43aa3eae4d69da2b76e413e10123c2c64","schema_version":"1.0","event_id":"sha256:d56b0e2198c2aa40b440a285311c04c43aa3eae4d69da2b76e413e10123c2c64"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CLSZR5EUPYGAUM4E5GFYJEQ32Z/bundle.json","state_url":"https://pith.science/pith/CLSZR5EUPYGAUM4E5GFYJEQ32Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CLSZR5EUPYGAUM4E5GFYJEQ32Z/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":"2026-05-29T14:53:19Z","links":{"resolver":"https://pith.science/pith/CLSZR5EUPYGAUM4E5GFYJEQ32Z","bundle":"https://pith.science/pith/CLSZR5EUPYGAUM4E5GFYJEQ32Z/bundle.json","state":"https://pith.science/pith/CLSZR5EUPYGAUM4E5GFYJEQ32Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CLSZR5EUPYGAUM4E5GFYJEQ32Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:CLSZR5EUPYGAUM4E5GFYJEQ32Z","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":"2d8613c4a8f73f2d73ac4bc69a2c81e97de0a255ea447a65c53bc9a5918d99b0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-04T03:54:06Z","title_canon_sha256":"37b9ed47c7878c49cefa9e6e4e4b92d820ef4ea7199cf43065863904ac038776"},"schema_version":"1.0","source":{"id":"1604.00733","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.00733","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"arxiv_version","alias_value":"1604.00733v4","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.00733","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"pith_short_12","alias_value":"CLSZR5EUPYGA","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CLSZR5EUPYGAUM4E","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CLSZR5EU","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:d56b0e2198c2aa40b440a285311c04c43aa3eae4d69da2b76e413e10123c2c64","target":"graph","created_at":"2026-05-17T23:48:53Z","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":"Szemer\\'edi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze--Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence ","authors_text":"Jacob Fox, L\\'aszl\\'o Mikl\\'os Lov\\'asz, Yufei Zhao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-04T03:54:06Z","title":"On regularity lemmas and their algorithmic applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.00733","kind":"arxiv","version":4},"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":"sha256:66b96bf69b5d57fa898f34366c18af6383a9497b57fc8954861b1179707ac790","target":"record","created_at":"2026-05-17T23:48:53Z","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":"2d8613c4a8f73f2d73ac4bc69a2c81e97de0a255ea447a65c53bc9a5918d99b0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-04T03:54:06Z","title_canon_sha256":"37b9ed47c7878c49cefa9e6e4e4b92d820ef4ea7199cf43065863904ac038776"},"schema_version":"1.0","source":{"id":"1604.00733","kind":"arxiv","version":4}},"canonical_sha256":"12e598f4947e0c0a3384e98b84921bd676fa247d66f537439cf94bfc9f0432bb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"12e598f4947e0c0a3384e98b84921bd676fa247d66f537439cf94bfc9f0432bb","first_computed_at":"2026-05-17T23:48:53.302919Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:48:53.302919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Vair9VBxxPWYh65SqcqKrbduf7CKnLiaduNfL31E0/85hsiumIQFlz4SRYnfjO/GVU6tkZry1Sj+xoiCi5VjDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:48:53.303434Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.00733","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:66b96bf69b5d57fa898f34366c18af6383a9497b57fc8954861b1179707ac790","sha256:d56b0e2198c2aa40b440a285311c04c43aa3eae4d69da2b76e413e10123c2c64"],"state_sha256":"13d06a47715774cdf6d49d5f0b09eac98fc62cc451e91dce1040f7b2b8ca1424"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8JujFozkWLLQNx45gzdrGcdSy0DW+AQsz2+XNQofDt9+zXnWLc9jZFd1YpzHq1G8iqmjFMW3eqbdosvoK0ejDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T14:53:19.126212Z","bundle_sha256":"ca57712b367ec2ba5b2e604c808ea76bde54613e050efd0652bdb3f0e4befd8f"}}