{"paper":{"title":"An approximate isoperimetric inequality for r-sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Ellis, Demetres Christofides, Peter Keevash","submitted_at":"2012-03-16T13:23:29Z","abstract_excerpt":"We prove a vertex-isoperimetric inequality for [n]^(r), the set of all r-element subsets of {1,2,...,n}, where x,y \\in [n]^(r) are adjacent if |x \\Delta y|=2. Namely, if \\mathcal{A} \\subset [n]^(r) with |\\mathcal{A}|=\\alpha {n \\choose r}, then the vertex-boundary b(\\mathcal{A}) satisfies |b(\\mathcal{A})| \\geq c\\sqrt{\\frac{n}{r(n-r)}} \\alpha(1-\\alpha) {n \\choose r}, where c is a positive absolute constant. For \\alpha bounded away from 0 and 1, this is sharp up to a constant factor (independent of n and r)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.3699","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"}