{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:O3B4VNUEPH3IQGKKWIMW6L65LT","short_pith_number":"pith:O3B4VNUE","schema_version":"1.0","canonical_sha256":"76c3cab68479f688194ab2196f2fdd5cdff0854b8eeec6755df3f48804942624","source":{"kind":"arxiv","id":"1201.1207","version":1},"attestation_state":"computed","paper":{"title":"A Statement in Combinatorics that is Independent of ZFC (an exposition)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Stephen Fenner, William Gasarch","submitted_at":"2012-01-05T16:02:50Z","abstract_excerpt":"It is known that, for any finite coloring of the naturals, there exists distinct naturals $e_1,e_2,e_3,e_4$ that are the same color such that $e_1+e_2=e_3+e_4$. Consider the following statement which we denote S: For every $\\aleph_0$-coloring of the reals there exists distinct reals $e_1,e_2,e_3,e_4$ such that $e_1+e_2=e_3+e_4$?} Is it true? Erdos showed that S is equivalent to the negation of the Continuum Hypothesis, and hence S is indepedent of ZFC. We give an exposition of his proof and some modern observations about results of this sort."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1201.1207","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-05T16:02:50Z","cross_cats_sorted":[],"title_canon_sha256":"7f8fc65c8c116c5b96f73264fbcf4c524f03fdae2edb5361d242cb5f9cba0370","abstract_canon_sha256":"705376ca21d533b06c42ad69d85ab0ea05f20edb8c3b6f5e6ae1868dace903b9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:05.715205Z","signature_b64":"khsFpEu1wjZn12LyvtibwCaOg7HVlFTeWGYvP+otU0uhirSr9XRWhiVUqwNmjxTbnf0KnQ279gAnSpbyWO2mDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76c3cab68479f688194ab2196f2fdd5cdff0854b8eeec6755df3f48804942624","last_reissued_at":"2026-05-18T04:05:05.714747Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:05.714747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Statement in Combinatorics that is Independent of ZFC (an exposition)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Stephen Fenner, William Gasarch","submitted_at":"2012-01-05T16:02:50Z","abstract_excerpt":"It is known that, for any finite coloring of the naturals, there exists distinct naturals $e_1,e_2,e_3,e_4$ that are the same color such that $e_1+e_2=e_3+e_4$. Consider the following statement which we denote S: For every $\\aleph_0$-coloring of the reals there exists distinct reals $e_1,e_2,e_3,e_4$ such that $e_1+e_2=e_3+e_4$?} Is it true? Erdos showed that S is equivalent to the negation of the Continuum Hypothesis, and hence S is indepedent of ZFC. We give an exposition of his proof and some modern observations about results of this sort."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.1207","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1201.1207","created_at":"2026-05-18T04:05:05.714817+00:00"},{"alias_kind":"arxiv_version","alias_value":"1201.1207v1","created_at":"2026-05-18T04:05:05.714817+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.1207","created_at":"2026-05-18T04:05:05.714817+00:00"},{"alias_kind":"pith_short_12","alias_value":"O3B4VNUEPH3I","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_16","alias_value":"O3B4VNUEPH3IQGKK","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_8","alias_value":"O3B4VNUE","created_at":"2026-05-18T12:27:16.716162+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/O3B4VNUEPH3IQGKKWIMW6L65LT","json":"https://pith.science/pith/O3B4VNUEPH3IQGKKWIMW6L65LT.json","graph_json":"https://pith.science/api/pith-number/O3B4VNUEPH3IQGKKWIMW6L65LT/graph.json","events_json":"https://pith.science/api/pith-number/O3B4VNUEPH3IQGKKWIMW6L65LT/events.json","paper":"https://pith.science/paper/O3B4VNUE"},"agent_actions":{"view_html":"https://pith.science/pith/O3B4VNUEPH3IQGKKWIMW6L65LT","download_json":"https://pith.science/pith/O3B4VNUEPH3IQGKKWIMW6L65LT.json","view_paper":"https://pith.science/paper/O3B4VNUE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1201.1207&json=true","fetch_graph":"https://pith.science/api/pith-number/O3B4VNUEPH3IQGKKWIMW6L65LT/graph.json","fetch_events":"https://pith.science/api/pith-number/O3B4VNUEPH3IQGKKWIMW6L65LT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O3B4VNUEPH3IQGKKWIMW6L65LT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O3B4VNUEPH3IQGKKWIMW6L65LT/action/storage_attestation","attest_author":"https://pith.science/pith/O3B4VNUEPH3IQGKKWIMW6L65LT/action/author_attestation","sign_citation":"https://pith.science/pith/O3B4VNUEPH3IQGKKWIMW6L65LT/action/citation_signature","submit_replication":"https://pith.science/pith/O3B4VNUEPH3IQGKKWIMW6L65LT/action/replication_record"}},"created_at":"2026-05-18T04:05:05.714817+00:00","updated_at":"2026-05-18T04:05:05.714817+00:00"}