{
  "id": "1310.8179",
  "version": "v2",
  "published": "2013-10-30T14:52:58.000Z",
  "updated": "2013-11-27T10:34:08.000Z",
  "title": "Almost isoperimetric subsets of the discrete cube",
  "authors": [
    "David Ellis"
  ],
  "comment": "Typos corrected in Conjectures 12 and 13",
  "journal": "Combinatorics, Probability and Computing 20 (2011), 363-380",
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that a set $A \\subset \\{0,1\\}^{n}$ with edge-boundary of size at most $|A| (\\log_{2}(2^{n}/|A|) + \\epsilon)$ can be made into a subcube by at most $(2 \\epsilon/\\log_{2}(1/\\epsilon))|A|$ additions and deletions, provided $\\epsilon$ is less than an absolute constant. We deduce that if $A \\subset \\{0,1\\}^{n}$ has size $2^{t}$ for some $t \\in \\mathbb{N}$, and cannot be made into a subcube by fewer than $\\delta |A|$ additions and deletions, then its edge-boundary has size at least $|A| \\log_{2}(2^{n}/|A|) + |A| \\delta \\log_{2}(1/\\delta) = 2^{t}(n-t+\\delta \\log_{2}(1/\\delta))$, provided $\\delta$ is less than an absolute constant. This is sharp whenever $\\delta = 1/2^{j}$ for some $j \\in \\{1,2,\\ldots,t\\}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-27T10:34:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "discrete cube",
      "isoperimetric subsets",
      "absolute constant",
      "edge-boundary"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.8179E"
    }
  }
}