{
  "id": "2111.00858",
  "version": "v2",
  "published": "2021-11-01T11:56:31.000Z",
  "updated": "2022-08-01T09:51:34.000Z",
  "title": "Block avoiding point sequencings of partial Steiner systems",
  "authors": [
    "Daniel Horsley",
    "Padraig Ó Catháin"
  ],
  "comment": "9 pages, 0 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A partial $(n,k,t)_\\lambda$-system is a pair $(X,\\mathcal{B})$ where $X$ is an $n$-set of vertices and $\\mathcal{B}$ is a collection of $k$-subsets of $X$ called blocks such that each $t$-set of vertices is a subset of at most $\\lambda$ blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of $\\{0,\\ldots,n-1\\}$. A sequencing is $\\ell$-block avoiding or, more briefly, $\\ell$-good if no block is contained in a set of $\\ell$ vertices with consecutive labels. Here we give a short proof that, for fixed $k$, $t$ and $\\lambda$, any partial $(n,k,t)_\\lambda$-system has an $\\ell$-good sequencing for some $\\ell=\\Theta(n^{1/t})$ as $n$ becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case $k=t+1$ where results of Kostochka, Mubayi and Verstra\\\"{e}te show that the value of $\\ell$ cannot be increased beyond $\\Theta((n \\log n)^{1/t})$. A special case of our result shows that every partial Steiner triple system (partial $(n,3,2)_1$-system) has an $\\ell$-good sequencing for each positive integer $\\ell \\leq 0.0908\\,n^{1/2}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-08-01T09:51:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B05",
      "05B07"
    ],
    "keywords": [
      "block avoiding point sequencings",
      "partial steiner systems",
      "partial steiner triple system",
      "special case",
      "short proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}