{
  "id": "1205.1501",
  "version": "v2",
  "published": "2012-05-07T19:57:56.000Z",
  "updated": "2012-11-10T20:02:13.000Z",
  "title": "On diamond-free subposets of the Boolean lattice",
  "authors": [
    "Lucas Kramer",
    "Ryan R. Martin",
    "Michael Young"
  ],
  "comment": "23 pages, 10 figures Accepted to Journal of Combinatorial Theory, Series A",
  "doi": "10.1016/j.jcta.2012.11.002",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: $A\\subset B,C\\subset D$. A diamond-free family in the $n$-dimensional Boolean lattice is a subposet such that no four elements form a diamond. Note that elements $B$ and $C$ may or may not be related. There is a diamond-free family in the $n$-dimensional Boolean lattice of size $(2-o(1)){n\\choose\\lfloor n/2\\rfloor}$. In this paper, we prove that any diamond-free family in the $n$-dimensional Boolean lattice has size at most $(2.25+o(1)){n\\choose\\lfloor n/2\\rfloor}$. Furthermore, we show that the so-called Lubell function of a diamond-free family in the $n$-dimensional Boolean lattice is at most $2.25+o(1)$, which is asymptotically best possible.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-11-10T20:02:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07",
      "05D05",
      "05C35"
    ],
    "keywords": [
      "dimensional boolean lattice",
      "diamond-free subposets",
      "diamond-free family",
      "elements form",
      "distinct elements"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.1501K"
    }
  }
}