{
  "id": "1802.06143",
  "version": "v1",
  "published": "2018-02-16T22:02:22.000Z",
  "updated": "2018-02-16T22:02:22.000Z",
  "title": "On the Turán density of $\\{1, 3\\}$-Hypergraphs",
  "authors": [
    "Shuliang Bai",
    "Linyuan Lu"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we consider the Tur\\'an problems on $\\{1,3\\}$-hypergraphs. We prove that a $\\{1, 3\\}$-hypergraph is degenerate if and only if it's $H^{\\{1, 3\\}}_5$-colorable, where $H^{\\{1, 3\\}}_5$ is a hypergraph with vertex set $V=[5]$ and edge set $E=\\{\\{2\\}, \\{3\\}, \\{1, 2, 4\\}, \\{1, 3, 5\\}, \\{1, 4, 5\\}\\}.$ Using this result, we further prove that for any finite set $R$ of distinct positive integers, except the case $R=\\{1, 2\\}$, there always exist non-trivial degenerate $R$-graphs. We also compute the Tur\\'an densities of some small $\\{1,3\\}$-hypergraphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-16T22:02:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C65"
    ],
    "keywords": [
      "hypergraph",
      "turán density",
      "turan problems",
      "non-trivial degenerate",
      "edge set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}