{"paper":{"title":"A refinement of the Cameron-Erd\\H{o}s Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"J\\'ozsef Balogh, Noga Alon, Robert Morris, Wojciech Samotij","submitted_at":"2012-02-23T14:54:40Z","abstract_excerpt":"In this paper we study sum-free subsets of the set $\\{1,...,n\\}$, that is, subsets of the first $n$ positive integers which contain no solution to the equation $x + y = z$. Cameron and Erd\\H{o}s conjectured in 1990 that the number of such sets is $O(2^{n/2})$. This conjecture was confirmed by Green and, independently, by Sapozhenko. Here we prove a refined version of their theorem, by showing that the number of sum-free subsets of $[n]$ of size $m$ is $2^{O(n/m)} {\\lceil n/2 \\rceil \\choose m}$, for every $1 \\le m \\le \\lceil n/2 \\rceil$. For $m \\ge \\sqrt{n}$, this result is sharp up to the cons"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.5200","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"}