{
  "id": "1207.2923",
  "version": "v1",
  "published": "2012-07-12T11:12:21.000Z",
  "updated": "2012-07-12T11:12:21.000Z",
  "title": "Families that remain $k$-Sperner even after omitting an element of their ground set",
  "authors": [
    "Balazs Patkos"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A family $\\cF\\subseteq 2^{[n]}$ of sets is said to be $l$-trace $k$-Sperner if for any $l$-subset $L \\subset [n]$ the family $\\cF|_L=\\{F|_L:F \\in \\cF\\}=\\{F \\cap L: F \\in \\cF\\}$ is $k$-Sperner, i.e. does not contain any chain of length $k+1$. The maximum size that an $l$-trace $k$-Sperner family $\\cF \\subseteq 2^{[n]}$ can have is denoted by $f(n,k,l)$. For pairs of integers $l<k$, if in a family $\\cG$ every pair of sets satisfies $||G_1|-|G_2||<k-l$, then $\\cG$ possesses the $(n-l)$-trace $k$-Sperner property. Among such families, the largest one is $\\cF_0=\\{F\\in 2^{[n]}: \\lfloor \\frac{n-(k-l)}{2}\\rfloor+1 \\le |F| \\le \\lfloor \\frac{n-(k-l)}{2}\\rfloor +k-l\\}$ and also $\\cF'_0=\\{F\\in 2^{[n]}: \\lfloor \\frac{n-(k-l)}{2}\\rfloor \\le |F| \\le \\lfloor \\frac{n-(k-l)}{2}\\rfloor +k-l-1\\}$ if $n-(k-l)$ is even. In an earlier paper, we proved that this is asymptotically optimal for all pair of integers $l<k$, i.e. $f(n,k,n-l)=(1+o(1))|\\cF_0|$. In this paper we consider the case when $l=1$, $k\\ge 2$, and prove that $f(n,k,n-1)=|\\cF_0|$ provided $n$ is large enough. We also prove that the unique $(n-1)$-trace $k$-Sperner family with size $f(n,k,n-1)$ is $\\cF_0$ and also $\\cF'_0$ when $n+k$ is odd.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-07-12T11:12:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "ground set",
      "sperner property",
      "sperner family",
      "sets satisfies",
      "earlier paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.2923P"
    }
  }
}