{
  "id": "1908.06333",
  "version": "v1",
  "published": "2019-08-17T18:47:55.000Z",
  "updated": "2019-08-17T18:47:55.000Z",
  "title": "Asymptotic enumeration of linear hypergraphs with given number of vertices and edges",
  "authors": [
    "Brendan D. McKay",
    "Fang Tian"
  ],
  "comment": "Submitted in January 2019",
  "categories": [
    "math.CO"
  ],
  "abstract": "For $n\\geq 3$, let $r=r(n)\\geq 3$ be an integer. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear $r$-uniform hypergraphs on $n\\to\\infty$ vertices is determined asymptotically when the number of edges is $m(n)=o(r^{-3}n^{ \\frac32})$. As one application, we find the probability of linearity for the independent-edge model of random $r$-uniform hypergraph when the expected number of edges is $o(r^{-3}n^{ \\frac32})$. We also find the probability that a random $r$-uniform linear hypergraph with a given number of edges contains a given subhypergraph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-17T18:47:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "05C80",
      "05A16"
    ],
    "keywords": [
      "asymptotic enumeration",
      "uniform hypergraph",
      "uniform linear hypergraph",
      "independent-edge model",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}