{
  "id": "1111.4636",
  "version": "v3",
  "published": "2011-11-20T14:23:31.000Z",
  "updated": "2017-12-01T15:12:54.000Z",
  "title": "A note on traces of set families",
  "authors": [
    "Balazs Patkos"
  ],
  "journal": "Moscow Journal of Combinatorics and Number Theory, 2 (2012) 47-55",
  "categories": [
    "math.CO"
  ],
  "abstract": "A family of sets $\\mathcal{F} \\subseteq 2^{[n]}$ is defined to be $l$-trace $k$-Sperner if for any $l$-subset $L$ of $[n]$ the family of traces $\\mathcal{F}|_L=\\{F \\cap L: F \\in \\mathcal{F}\\}$ does not contain any chain of length $k+1$. In this paper we prove that for any positive integers $l',k$ with $l'<k$ if $\\mathcal{F}$ is $(n-l')$-trace $k$-Sperner, then $|\\mathcal{F}| \\le (k-l'+o(1))\\binom{n}{\\lfloor n/2\\rfloor}$ and this bound is asymptotically tight.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-01-08T20:19:16.000Z",
      "abstract": "A family of sets $\\cF \\subseteq 2^{[n]}$ is defined to be $l$-trace $k$-Sperner if for any $l$-subset $L$ of $[n]$ the family of traces $\\cF|_L=\\{F \\cap L: F \\in \\cF\\}$ does not contain any chain of length $k+1$. In this paper we prove that for any positive integers $l',k$ with $l'<k$ if $\\cF$ is $(n-l')$-trace $k$-Sperner, then $|\\cF| \\le (k-l'+o(1))\\binom{n}{\\lfloor n/2\\rfloor}$ and this bound is asymptotically tight.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2017-12-01T15:12:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "set families",
      "positive integers",
      "asymptotically tight"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.4636P"
    }
  }
}