{"paper":{"title":"The largest projective cube-free subsets of $\\mathbb{Z}_{2^n}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Zsolt Wagner, Jason Long","submitted_at":"2018-10-02T13:20:45Z","abstract_excerpt":"In the Boolean lattice, Sperner's, Erd\\H{o}s's, Kleitman's and Samotij's theorems state that families that do not contain many chains must have a very specific layered structure. We show that if instead of $\\mathbb{Z}_2^n$ we work in $\\mathbb{Z}_{2^n}$, several analogous statements hold if one replaces the word $k$-chain by projective cube of dimension $2^{k-1}$.\n  We say that $B_d$ is a projective cube of dimension $d$ if there are numbers $a_1, a_2, \\ldots, a_d$ such that $$B_d = \\left\\{\\sum_{i\\in I} a_i \\bigg\\rvert \\emptyset \\neq I\\subseteq [d]\\right\\}.$$\n  As an analog of Sperner's and Erd"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.01225","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"}