{"paper":{"title":"On solution-free sets of integers II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Andrew Treglown, Robert Hancock","submitted_at":"2016-11-25T15:54:20Z","abstract_excerpt":"Given a linear equation $\\mathcal{L}$, a set $A \\subseteq [n]$ is $\\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\\mathcal{L}$. We determine the precise size of the largest $\\mathcal{L}$-free subset of $[n]$ for several general classes of linear equations $\\mathcal{L}$ of the form $px+qy=rz$ for fixed $p,q,r \\in \\mathbb N$ where $p \\geq q \\geq r$. Further, for all such linear equations $\\mathcal{L}$, we give an upper bound on the number of maximal $\\mathcal{L}$-free subsets of $[n]$. In the case when $p=q\\geq 2$ and $r=1$ this bound is exact up to an error term in th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08498","kind":"arxiv","version":2},"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"}