{
  "id": "1108.5603",
  "version": "v1",
  "published": "2011-08-29T15:26:16.000Z",
  "updated": "2011-08-29T15:26:16.000Z",
  "title": "Probably Intersecting Families are Not Nested",
  "authors": [
    "Paul A. Russell",
    "Mark Walters"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "It is well known that an intersecting family of subsets of an n-element set can contain at most 2^(n-1) sets. It is natural to wonder how `close' to intersecting a family of size greater than 2^(n-1) can be. Katona, Katona and Katona introduced the idea of a `most probably intersecting family.' Suppose that X is a family and that 0<p<1. Let X(p) be the (random) family formed by selecting each set in X independently with probability p. A family X is `most probably intersecting' if it maximises the probability that X(p) is intersecting over all families of size |X|. Katona, Katona and Katona conjectured that there is a nested sequence consisting of most probably intersecting families of every possible size. We show that this conjecture is false for every value of p provided that n is sufficiently large.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-29T15:26:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "probably intersecting family",
      "n-element set",
      "probability",
      "conjecture",
      "nested sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.5603R"
    }
  }
}