{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:ROH5HX6HT32LYE7MQ6G5DKMBGI","short_pith_number":"pith:ROH5HX6H","schema_version":"1.0","canonical_sha256":"8b8fd3dfc79ef4bc13ec878dd1a981321be4668233540c229383f6468a718b3f","source":{"kind":"arxiv","id":"1401.6390","version":1},"attestation_state":"computed","paper":{"title":"F{\\o}lner sequences and sum-free sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Sean Eberhard","submitted_at":"2014-01-24T16:18:44Z","abstract_excerpt":"Erd\\H{o}s showed that every set of $n$ positive integers contains a subset of size at least $n/(k+1)$ containing no solutions to $x_1 + \\cdots + x_k = y$. We prove that the constant $1/(k+1)$ here is best possible by showing that if $(F_m)$ is a multiplicative F{\\o}lner sequence in $\\mathbf{N}$ then $F_m$ has no $k$-sum-free subset of size greater than $(1/(k+1)+o(1))|F_m|$. This provides a new proof and a generalisation of a recent theorem of Eberhard, Green, and Manners."},"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":"1401.6390","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-01-24T16:18:44Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"2e7de7e97ff9865523394f3250aca30897a18c6872b36781c060cac55d2b52fd","abstract_canon_sha256":"5c1c33bed4536e78b75570eccfc91eccf41d0766264567b0ba173bc354036040"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:31:11.987827Z","signature_b64":"C5mCp5yImyMALjQcek1IRrURVCglFUnsqSyqZTIgmuAR1RrOyzZ9NvqRHEXJD027vFOl1s0XvA1bUuGG5xLdCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b8fd3dfc79ef4bc13ec878dd1a981321be4668233540c229383f6468a718b3f","last_reissued_at":"2026-05-18T02:31:11.987197Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:31:11.987197Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"F{\\o}lner sequences and sum-free sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Sean Eberhard","submitted_at":"2014-01-24T16:18:44Z","abstract_excerpt":"Erd\\H{o}s showed that every set of $n$ positive integers contains a subset of size at least $n/(k+1)$ containing no solutions to $x_1 + \\cdots + x_k = y$. We prove that the constant $1/(k+1)$ here is best possible by showing that if $(F_m)$ is a multiplicative F{\\o}lner sequence in $\\mathbf{N}$ then $F_m$ has no $k$-sum-free subset of size greater than $(1/(k+1)+o(1))|F_m|$. This provides a new proof and a generalisation of a recent theorem of Eberhard, Green, and Manners."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.6390","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":"1401.6390","created_at":"2026-05-18T02:31:11.987293+00:00"},{"alias_kind":"arxiv_version","alias_value":"1401.6390v1","created_at":"2026-05-18T02:31:11.987293+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.6390","created_at":"2026-05-18T02:31:11.987293+00:00"},{"alias_kind":"pith_short_12","alias_value":"ROH5HX6HT32L","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_16","alias_value":"ROH5HX6HT32LYE7M","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_8","alias_value":"ROH5HX6H","created_at":"2026-05-18T12:28:46.137349+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/ROH5HX6HT32LYE7MQ6G5DKMBGI","json":"https://pith.science/pith/ROH5HX6HT32LYE7MQ6G5DKMBGI.json","graph_json":"https://pith.science/api/pith-number/ROH5HX6HT32LYE7MQ6G5DKMBGI/graph.json","events_json":"https://pith.science/api/pith-number/ROH5HX6HT32LYE7MQ6G5DKMBGI/events.json","paper":"https://pith.science/paper/ROH5HX6H"},"agent_actions":{"view_html":"https://pith.science/pith/ROH5HX6HT32LYE7MQ6G5DKMBGI","download_json":"https://pith.science/pith/ROH5HX6HT32LYE7MQ6G5DKMBGI.json","view_paper":"https://pith.science/paper/ROH5HX6H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1401.6390&json=true","fetch_graph":"https://pith.science/api/pith-number/ROH5HX6HT32LYE7MQ6G5DKMBGI/graph.json","fetch_events":"https://pith.science/api/pith-number/ROH5HX6HT32LYE7MQ6G5DKMBGI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ROH5HX6HT32LYE7MQ6G5DKMBGI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ROH5HX6HT32LYE7MQ6G5DKMBGI/action/storage_attestation","attest_author":"https://pith.science/pith/ROH5HX6HT32LYE7MQ6G5DKMBGI/action/author_attestation","sign_citation":"https://pith.science/pith/ROH5HX6HT32LYE7MQ6G5DKMBGI/action/citation_signature","submit_replication":"https://pith.science/pith/ROH5HX6HT32LYE7MQ6G5DKMBGI/action/replication_record"}},"created_at":"2026-05-18T02:31:11.987293+00:00","updated_at":"2026-05-18T02:31:11.987293+00:00"}