{"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"}