{
  "id": "1604.02160",
  "version": "v1",
  "published": "2016-04-07T20:08:02.000Z",
  "updated": "2016-04-07T20:08:02.000Z",
  "title": "Stability versions of Erdős-Ko-Rado type theorems, via isoperimetry",
  "authors": [
    "David Ellis",
    "Nathan Keller",
    "Noam Lifshitz"
  ],
  "comment": "70 pages, one Appendix",
  "categories": [
    "math.CO"
  ],
  "abstract": "Erd\\H{o}s-Ko-Rado (EKR) type theorems give upper bounds on the sizes of families of sets, under various intersection requirements on the sets in the family. `Stability' versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in an appropriate sense) to a maximum-sized family. In this paper, we present an approach to obtaining stability versions of EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube. We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede-Khachatrian theorem on $t$-intersecting families of $k$-element subsets of $\\{1,2,\\ldots,n\\}$ (for $k < \\frac{n}{t+1}$), and to show that, somewhat surprisingly, the results hold when the `intersection' requirement is replaced by a much weaker requirement. We also obtain stability versions of several more recent EKR-type results, including Frankl's recent result on the Erd\\H{o}s matching conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-07T20:08:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "erdős-ko-rado type theorems",
      "isoperimetry",
      "tight stability versions",
      "intersection requirements",
      "theorems assert"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 70,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160402160E"
    }
  }
}