{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:CHL5G5SNUDDBHNJ7HCHB6AILCU","short_pith_number":"pith:CHL5G5SN","schema_version":"1.0","canonical_sha256":"11d7d3764da0c613b53f388e1f010b1539b37e587ca33c1d1170dc235f428ee4","source":{"kind":"arxiv","id":"1507.03764","version":3},"attestation_state":"computed","paper":{"title":"The number of additive triples in subsets of abelian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benny Sudakov, Wojciech Samotij","submitted_at":"2015-07-14T08:36:24Z","abstract_excerpt":"A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of how large the largest sum-free subset of a given abelian group is had started more than thirty years before it was finally resolved by Green and Ruzsa a decade ago. We address the following more general question. Suppose that a set $A$ of elements of an abelian group $G$ has cardinality $a$. How many Schur triples must $A$ contain? Moreover, which sets of $a$ elements of $G$ have the smallest number of Schur triples? In this paper, we ans"},"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":"1507.03764","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-14T08:36:24Z","cross_cats_sorted":[],"title_canon_sha256":"4fa59ede43e84c666c25df54b2b8d5fd0c1dcfecf024fce6a0a906dfac19237d","abstract_canon_sha256":"04138afce7a99c407dd4596c3777a8833a1be247381630f479c4be1ed835a235"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:45.937534Z","signature_b64":"8hD6KhrLT7qJ7SzxZ+3wgD4U8IctPt6B3774tMLsIAevmKMJj7mKfKygn1iLaUjPh3xrE7spZWFAf0jTs9p2Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11d7d3764da0c613b53f388e1f010b1539b37e587ca33c1d1170dc235f428ee4","last_reissued_at":"2026-05-18T01:10:45.937027Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:45.937027Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The number of additive triples in subsets of abelian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benny Sudakov, Wojciech Samotij","submitted_at":"2015-07-14T08:36:24Z","abstract_excerpt":"A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of how large the largest sum-free subset of a given abelian group is had started more than thirty years before it was finally resolved by Green and Ruzsa a decade ago. We address the following more general question. Suppose that a set $A$ of elements of an abelian group $G$ has cardinality $a$. How many Schur triples must $A$ contain? Moreover, which sets of $a$ elements of $G$ have the smallest number of Schur triples? In this paper, we ans"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03764","kind":"arxiv","version":3},"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":"1507.03764","created_at":"2026-05-18T01:10:45.937100+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.03764v3","created_at":"2026-05-18T01:10:45.937100+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03764","created_at":"2026-05-18T01:10:45.937100+00:00"},{"alias_kind":"pith_short_12","alias_value":"CHL5G5SNUDDB","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_16","alias_value":"CHL5G5SNUDDBHNJ7","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_8","alias_value":"CHL5G5SN","created_at":"2026-05-18T12:29:14.074870+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/CHL5G5SNUDDBHNJ7HCHB6AILCU","json":"https://pith.science/pith/CHL5G5SNUDDBHNJ7HCHB6AILCU.json","graph_json":"https://pith.science/api/pith-number/CHL5G5SNUDDBHNJ7HCHB6AILCU/graph.json","events_json":"https://pith.science/api/pith-number/CHL5G5SNUDDBHNJ7HCHB6AILCU/events.json","paper":"https://pith.science/paper/CHL5G5SN"},"agent_actions":{"view_html":"https://pith.science/pith/CHL5G5SNUDDBHNJ7HCHB6AILCU","download_json":"https://pith.science/pith/CHL5G5SNUDDBHNJ7HCHB6AILCU.json","view_paper":"https://pith.science/paper/CHL5G5SN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.03764&json=true","fetch_graph":"https://pith.science/api/pith-number/CHL5G5SNUDDBHNJ7HCHB6AILCU/graph.json","fetch_events":"https://pith.science/api/pith-number/CHL5G5SNUDDBHNJ7HCHB6AILCU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CHL5G5SNUDDBHNJ7HCHB6AILCU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CHL5G5SNUDDBHNJ7HCHB6AILCU/action/storage_attestation","attest_author":"https://pith.science/pith/CHL5G5SNUDDBHNJ7HCHB6AILCU/action/author_attestation","sign_citation":"https://pith.science/pith/CHL5G5SNUDDBHNJ7HCHB6AILCU/action/citation_signature","submit_replication":"https://pith.science/pith/CHL5G5SNUDDBHNJ7HCHB6AILCU/action/replication_record"}},"created_at":"2026-05-18T01:10:45.937100+00:00","updated_at":"2026-05-18T01:10:45.937100+00:00"}