{
  "id": "0912.5039",
  "version": "v3",
  "published": "2009-12-26T20:27:08.000Z",
  "updated": "2016-05-21T14:43:22.000Z",
  "title": "$Q_2$-free families in the Boolean lattice",
  "authors": [
    "Maria Axenovich",
    "Jacob Manske",
    "Ryan R. Martin"
  ],
  "comment": "18 pages, 2 figures",
  "journal": "Order 29(1) (2012), 177--191",
  "doi": "10.1007/s11083-011-9206-4",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a family $\\mathcal{F}$ of subsets of [n]=\\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \\leq ex(n, Q_2)\\leq 2.283261N +o(N), $ where $N = \\binom{n}{\\lfloor n/2 \\rfloor}$. We also prove that the largest $Q_2$-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-02-15T21:11:51.000Z",
      "journal": null,
      "doi": null,
      "authors": [
        "Maria Axenovich",
        "Jacob Manske",
        "Ryan Martin"
      ]
    },
    {
      "version": "v3",
      "updated": "2016-05-21T14:43:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "06A07"
    ],
    "keywords": [
      "boolean lattice",
      "free family",
      "subposet isomorphic",
      "distinct elements",
      "20711n members"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.5039A"
    }
  }
}