{"paper":{"title":"Unrolling residues to avoid progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Linyuan Lu, Ron Graham, Steve Butler","submitted_at":"2012-09-12T18:40:00Z","abstract_excerpt":"We consider the problem of coloring $[n]={1,2,...,n}$ with $r$ colors to minimize the number of monochromatic $k$ term arithmetic progressions (or $k$-APs for short). We show how to extend colorings of $\\mathbb{Z}_m$ which avoid nontrivial $k$-APs to colorings of $[n]$ by an unrolling process. In particular, by using residues to color $\\mathbb{Z}_m$ we produce the best known colorings for minimizing the number of monochromatic $k$-APs for coloring with $r$ colors for several small values of $r$ and $k$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2687","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"}