{"paper":{"title":"Sets of natural numbers with proscribed subsets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Kevin O'Bryant","submitted_at":"2014-10-18T02:34:41Z","abstract_excerpt":"Fix $A$, a family of subsets of natural numbers, and let $G_A(n)$ be the maximum cardinality of a subset of $\\{1,2,..., n\\}$ that does not have any subset in $A$. We consider the general problem of giving upper bounds on $G_A(n)$ and give some new upper bounds on some families that are closed under dilation. Specific examples include sets that do not contain any geometric progression of length $k$ with integer ratio, sets that do not contain any geometric progression of length $k$ with rational ratio, and sets of integers that do not contain multiplicative squares, i.e., nontrivial sets of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.4900","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"}