{
  "id": "1703.05427",
  "version": "v1",
  "published": "2017-03-15T23:40:25.000Z",
  "updated": "2017-03-15T23:40:25.000Z",
  "title": "Families in posets minimizing the number of comparable pairs",
  "authors": [
    "Jozsef Balogh",
    "Sarka Petrickova",
    "Adam Zsolt Wagner"
  ],
  "comment": "20 pages, 6 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a poset $P$ we say a family $\\mathcal{F}\\subseteq P$ is centered if it is obtained by `taking sets as close to the middle layer as possible'. A poset $P$ is said to have the centeredness property if for any $M$, among all families of size $M$ in $P$, centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice $\\{0,1\\}^n$ has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset $\\{0,1,\\ldots,k\\}^n$ also has the centeredness property, provided $n$ is sufficiently large compared to $k$. We show that this conjecture is false for all $k\\geq 2$ and investigate the range of $M$ for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of $\\mathbb{F}_q^n$ has the centeredness property. Several open questions are also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-15T23:40:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "comparable pairs",
      "centeredness property",
      "posets minimizing",
      "middle layer",
      "minimum number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}