{"paper":{"title":"On solution-free sets of integers","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":"2016-07-28T11:07:18Z","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 $\\mathcal{L}$. In this paper we consider the following three general questions:\n  (i) What is the size of the largest $\\mathcal{L}$-free subset of $[n]$?\n  (ii) How many $\\mathcal{L}$-free subsets of $[n]$ are there?\n  (iii) How many maximal $\\mathcal{L}$-free subsets of $[n]$ are there?\n  We completely resolve (i) in the case when $\\mathcal{L}$ is the equation $px+qy=z$ for fixed $p,q\\in \\mathbb N$ where $p\\geq 2$. Further, up to a multiplicative constant"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.08399","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"}