{
  "id": "2310.01822",
  "version": "v1",
  "published": "2023-10-03T06:37:59.000Z",
  "updated": "2023-10-03T06:37:59.000Z",
  "title": "Simplicial Turán problems",
  "authors": [
    "David Conlon",
    "Simón Piga",
    "Bjarne Schülke"
  ],
  "comment": "22 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A simplicial complex $H$ consists of a pair of sets $(V,E)$ where $V$ is a set of vertices and $E\\subseteq\\mathscr{P}(V)$ is a collection of subsets of $V$ closed under taking subsets. Given a simplicial complex $F$ and $n\\in \\mathbb N$, the extremal number $\\text{ex}(n,F)$ is the maximum number of edges that a simplicial complex on $n$ vertices can have without containing a copy of $F$. We initiate the systematic study of extremal numbers in this context by asymptotically determining the extremal numbers of several natural simplicial complexes. In particular, we asymptotically determine the extremal number of a simplicial complex for which the extremal example has more than one incomplete layer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-03T06:37:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "05C35",
      "05D05",
      "05D99"
    ],
    "keywords": [
      "simplicial turán problems",
      "extremal number",
      "natural simplicial complexes",
      "maximum number",
      "incomplete layer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}