{"paper":{"title":"On the structure of subsets of the discrete cube with small edge boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Ellis, Nathan Keller, Noam Lifshitz","submitted_at":"2016-12-20T14:36:29Z","abstract_excerpt":"The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of $\\{0,1\\}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on $\\{0,1\\}^n$. We show that for any $m$-element subset $\\mathcal{F} \\subset \\{0,1\\}^n$ and any integer $l$, if the edge boundary of $\\mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $\\mathcal{G} \\subset \\{0,1\\}^n$ such that $|\\mathcal{F} \\Delta \\mathcal{G}| "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.06680","kind":"arxiv","version":4},"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"}