{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:VXEDFS2M5SGIKGVRNWUNMUA7PA","short_pith_number":"pith:VXEDFS2M","schema_version":"1.0","canonical_sha256":"adc832cb4cec8c851ab16da8d6501f78132224f3a01d6ef6d7f4fcfe9ee3af57","source":{"kind":"arxiv","id":"1608.05740","version":3},"attestation_state":"computed","paper":{"title":"Proof of a Conjecture of Kleinberg-Sawin-Speyer","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Luke Pebody","submitted_at":"2016-08-19T21:19:40Z","abstract_excerpt":"In Ellenberg and Gijswijt's groundbreaking work, the authors show that a subset of $\\mathbb{Z}_3^{n}$ with no arithmetic progression of length 3 must be of size at most $2.755^n$ (no prior upper bound was known of $(3-\\epsilon)^n)$), and provide for any prime $p$ a value $\\lambda_p<p$ such that any subset of $\\mathbb{Z}_p^{n}$ with no arithmetic progression of length 3 must be of size at most $\\lambda_p^n$.\n  Blasiak et al showed that the same bounds apply to tri-coloured sum-free sets, which are triples $\\{(a_i,b_i,c_i):a_i,b_i,c_i\\in\\mathbb{Z}_p^{n}\\}$ with $a_i+b_j+c_k=0$ if and only if $i="},"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":"1608.05740","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2016-08-19T21:19:40Z","cross_cats_sorted":[],"title_canon_sha256":"6f524545fcd021c74b0075ad4cf57ab1d7aa77f24f3439d203cfab51e1484365","abstract_canon_sha256":"6fa385413bf6735bd5bda7cd3bb6480dcffcd0800e3b12826af378872fe7f2c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:28.334040Z","signature_b64":"NiXpga+gZ6tRN9RUpj52aE2HGXVNEKzb/2Yansj/ieLHygcKp1ncRgogmfGaGvnFDazBAd/UaFfeu7Ja0bIWAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"adc832cb4cec8c851ab16da8d6501f78132224f3a01d6ef6d7f4fcfe9ee3af57","last_reissued_at":"2026-05-18T00:11:28.333687Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:28.333687Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of a Conjecture of Kleinberg-Sawin-Speyer","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Luke Pebody","submitted_at":"2016-08-19T21:19:40Z","abstract_excerpt":"In Ellenberg and Gijswijt's groundbreaking work, the authors show that a subset of $\\mathbb{Z}_3^{n}$ with no arithmetic progression of length 3 must be of size at most $2.755^n$ (no prior upper bound was known of $(3-\\epsilon)^n)$), and provide for any prime $p$ a value $\\lambda_p<p$ such that any subset of $\\mathbb{Z}_p^{n}$ with no arithmetic progression of length 3 must be of size at most $\\lambda_p^n$.\n  Blasiak et al showed that the same bounds apply to tri-coloured sum-free sets, which are triples $\\{(a_i,b_i,c_i):a_i,b_i,c_i\\in\\mathbb{Z}_p^{n}\\}$ with $a_i+b_j+c_k=0$ if and only if $i="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.05740","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":"1608.05740","created_at":"2026-05-18T00:11:28.333742+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.05740v3","created_at":"2026-05-18T00:11:28.333742+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.05740","created_at":"2026-05-18T00:11:28.333742+00:00"},{"alias_kind":"pith_short_12","alias_value":"VXEDFS2M5SGI","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_16","alias_value":"VXEDFS2M5SGIKGVR","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_8","alias_value":"VXEDFS2M","created_at":"2026-05-18T12:30:48.956258+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/VXEDFS2M5SGIKGVRNWUNMUA7PA","json":"https://pith.science/pith/VXEDFS2M5SGIKGVRNWUNMUA7PA.json","graph_json":"https://pith.science/api/pith-number/VXEDFS2M5SGIKGVRNWUNMUA7PA/graph.json","events_json":"https://pith.science/api/pith-number/VXEDFS2M5SGIKGVRNWUNMUA7PA/events.json","paper":"https://pith.science/paper/VXEDFS2M"},"agent_actions":{"view_html":"https://pith.science/pith/VXEDFS2M5SGIKGVRNWUNMUA7PA","download_json":"https://pith.science/pith/VXEDFS2M5SGIKGVRNWUNMUA7PA.json","view_paper":"https://pith.science/paper/VXEDFS2M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.05740&json=true","fetch_graph":"https://pith.science/api/pith-number/VXEDFS2M5SGIKGVRNWUNMUA7PA/graph.json","fetch_events":"https://pith.science/api/pith-number/VXEDFS2M5SGIKGVRNWUNMUA7PA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VXEDFS2M5SGIKGVRNWUNMUA7PA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VXEDFS2M5SGIKGVRNWUNMUA7PA/action/storage_attestation","attest_author":"https://pith.science/pith/VXEDFS2M5SGIKGVRNWUNMUA7PA/action/author_attestation","sign_citation":"https://pith.science/pith/VXEDFS2M5SGIKGVRNWUNMUA7PA/action/citation_signature","submit_replication":"https://pith.science/pith/VXEDFS2M5SGIKGVRNWUNMUA7PA/action/replication_record"}},"created_at":"2026-05-18T00:11:28.333742+00:00","updated_at":"2026-05-18T00:11:28.333742+00:00"}