{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:KLENXZB2W6YNN7WEXILUVW4MHN","short_pith_number":"pith:KLENXZB2","canonical_record":{"source":{"id":"1610.06359","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-20T11:16:59Z","cross_cats_sorted":[],"title_canon_sha256":"83495110fbcee4e71d1060d3bfce7932b9cecd7d47ccf9e24f7fa008395ddb0b","abstract_canon_sha256":"d008dc5251162ac5862f347d20230e14d4b31d98fa9f17bdb9f05b1de931f584"},"schema_version":"1.0"},"canonical_sha256":"52c8dbe43ab7b0d6fec4ba174adb8c3b5ce93222b9a453d9453e72a2a91ef155","source":{"kind":"arxiv","id":"1610.06359","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.06359","created_at":"2026-05-18T00:26:18Z"},{"alias_kind":"arxiv_version","alias_value":"1610.06359v2","created_at":"2026-05-18T00:26:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.06359","created_at":"2026-05-18T00:26:18Z"},{"alias_kind":"pith_short_12","alias_value":"KLENXZB2W6YN","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KLENXZB2W6YNN7WE","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KLENXZB2","created_at":"2026-05-18T12:30:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:KLENXZB2W6YNN7WEXILUVW4MHN","target":"record","payload":{"canonical_record":{"source":{"id":"1610.06359","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-20T11:16:59Z","cross_cats_sorted":[],"title_canon_sha256":"83495110fbcee4e71d1060d3bfce7932b9cecd7d47ccf9e24f7fa008395ddb0b","abstract_canon_sha256":"d008dc5251162ac5862f347d20230e14d4b31d98fa9f17bdb9f05b1de931f584"},"schema_version":"1.0"},"canonical_sha256":"52c8dbe43ab7b0d6fec4ba174adb8c3b5ce93222b9a453d9453e72a2a91ef155","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:18.491039Z","signature_b64":"sZH/BDsWQ67mXM+rqffIoflUQnPkj07NUoCLrCFt1iLEvBEXpdxsAaywK4dqAXr9POnc6dIWj8I6MvkUFolJBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52c8dbe43ab7b0d6fec4ba174adb8c3b5ce93222b9a453d9453e72a2a91ef155","last_reissued_at":"2026-05-18T00:26:18.490427Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:18.490427Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.06359","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:26:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mAvrGkL6PJYWq8Sm/EdqPkRBqrtaezaxW6VTeLDpD+wTZgoexqN9iFW37Y3kolEuZJfkhcgCkMTjjCia8rspAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:47:19.276041Z"},"content_sha256":"e799820893b4de87aed4bac8145e8624de15bfb2678e3968b59480a20e90da18","schema_version":"1.0","event_id":"sha256:e799820893b4de87aed4bac8145e8624de15bfb2678e3968b59480a20e90da18"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:KLENXZB2W6YNN7WEXILUVW4MHN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Discrepancy and large dense monochromatic subsets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guus Regts, Ross Kang, Viresh Patel","submitted_at":"2016-10-20T11:16:59Z","abstract_excerpt":"Erd\\H{o}s and Pach (1983) introduced the natural degree-based generalisations of Ramsey numbers, where instead of seeking large monochromatic cliques in a $2$-edge coloured complete graph, we seek monochromatic subgraphs of high minimum or average degree. Here we expand the study of these so-called quasi-Ramsey numbers in a few ways, in particular, to multiple colours and to uniform hypergraphs.\n  Quasi-Ramsey numbers are known to exhibit a certain unique phase transition and we show that this is also the case across the settings we consider. Our results depend on a density-biased notion of hy"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06359","kind":"arxiv","version":2},"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":"2026-05-18T00:26:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"joQ1Hw+71jklHlEHRWXhUfUyLG5SiBS8ppSYMKBje4qYiqeCKVTfwejRjmlTW9yr1Y4k2vLSR4HW5IvYMit7Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:47:19.276412Z"},"content_sha256":"a37bf66fd54a8fcdffe2bb0862594598dcbafa877d9b0cf705c83fa922c43fc6","schema_version":"1.0","event_id":"sha256:a37bf66fd54a8fcdffe2bb0862594598dcbafa877d9b0cf705c83fa922c43fc6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KLENXZB2W6YNN7WEXILUVW4MHN/bundle.json","state_url":"https://pith.science/pith/KLENXZB2W6YNN7WEXILUVW4MHN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KLENXZB2W6YNN7WEXILUVW4MHN/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":"2026-06-02T15:47:19Z","links":{"resolver":"https://pith.science/pith/KLENXZB2W6YNN7WEXILUVW4MHN","bundle":"https://pith.science/pith/KLENXZB2W6YNN7WEXILUVW4MHN/bundle.json","state":"https://pith.science/pith/KLENXZB2W6YNN7WEXILUVW4MHN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KLENXZB2W6YNN7WEXILUVW4MHN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:KLENXZB2W6YNN7WEXILUVW4MHN","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":"d008dc5251162ac5862f347d20230e14d4b31d98fa9f17bdb9f05b1de931f584","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-20T11:16:59Z","title_canon_sha256":"83495110fbcee4e71d1060d3bfce7932b9cecd7d47ccf9e24f7fa008395ddb0b"},"schema_version":"1.0","source":{"id":"1610.06359","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.06359","created_at":"2026-05-18T00:26:18Z"},{"alias_kind":"arxiv_version","alias_value":"1610.06359v2","created_at":"2026-05-18T00:26:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.06359","created_at":"2026-05-18T00:26:18Z"},{"alias_kind":"pith_short_12","alias_value":"KLENXZB2W6YN","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KLENXZB2W6YNN7WE","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KLENXZB2","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:a37bf66fd54a8fcdffe2bb0862594598dcbafa877d9b0cf705c83fa922c43fc6","target":"graph","created_at":"2026-05-18T00:26:18Z","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":"Erd\\H{o}s and Pach (1983) introduced the natural degree-based generalisations of Ramsey numbers, where instead of seeking large monochromatic cliques in a $2$-edge coloured complete graph, we seek monochromatic subgraphs of high minimum or average degree. Here we expand the study of these so-called quasi-Ramsey numbers in a few ways, in particular, to multiple colours and to uniform hypergraphs.\n  Quasi-Ramsey numbers are known to exhibit a certain unique phase transition and we show that this is also the case across the settings we consider. Our results depend on a density-biased notion of hy","authors_text":"Guus Regts, Ross Kang, Viresh Patel","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-20T11:16:59Z","title":"Discrepancy and large dense monochromatic subsets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06359","kind":"arxiv","version":2},"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:e799820893b4de87aed4bac8145e8624de15bfb2678e3968b59480a20e90da18","target":"record","created_at":"2026-05-18T00:26:18Z","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":"d008dc5251162ac5862f347d20230e14d4b31d98fa9f17bdb9f05b1de931f584","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-20T11:16:59Z","title_canon_sha256":"83495110fbcee4e71d1060d3bfce7932b9cecd7d47ccf9e24f7fa008395ddb0b"},"schema_version":"1.0","source":{"id":"1610.06359","kind":"arxiv","version":2}},"canonical_sha256":"52c8dbe43ab7b0d6fec4ba174adb8c3b5ce93222b9a453d9453e72a2a91ef155","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"52c8dbe43ab7b0d6fec4ba174adb8c3b5ce93222b9a453d9453e72a2a91ef155","first_computed_at":"2026-05-18T00:26:18.490427Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:18.490427Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sZH/BDsWQ67mXM+rqffIoflUQnPkj07NUoCLrCFt1iLEvBEXpdxsAaywK4dqAXr9POnc6dIWj8I6MvkUFolJBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:18.491039Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.06359","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e799820893b4de87aed4bac8145e8624de15bfb2678e3968b59480a20e90da18","sha256:a37bf66fd54a8fcdffe2bb0862594598dcbafa877d9b0cf705c83fa922c43fc6"],"state_sha256":"e4ff1973f2c15d8a72d8f1ff1248b50b70a4cae638189d00b99d28866c12f90e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vwPMzNpwrjPK1KakPQjty9Z9TwGJHMsYSjgyyt80H28DKGSD4Ig4+AqlD7LAiBk7/JQVcoWSIWW3BseFiYvoAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T15:47:19.278477Z","bundle_sha256":"7f49b74cf97e9187421a5c3c7d5a751746fd4db40bcbbb62851eb28df55c3f0d"}}