{
  "id": "2107.14798",
  "version": "v1",
  "published": "2021-07-30T17:54:48.000Z",
  "updated": "2021-07-30T17:54:48.000Z",
  "title": "Counting $r$-graphs without forbidden configurations",
  "authors": [
    "József Balogh",
    "Felix Christian Clemen",
    "Letícia Mattos"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "One of the major problems in combinatorics is to determine the number of $r$-uniform hypergraphs ($r$-graphs) on $n$ vertices which are free of certain forbidden structures. This problem dates back to the work of Erd\\H{o}s, Kleitman and Rothschild, who showed that the number of $K_r$-free graphs on $n$ vertices is $2^{\\text{ex}(n,K_r)+o(n^2)}$. Their work was later extended to forbidding graphs as induced subgraphs by Pr\\\"omel and Steger. Here, we consider one of the most basic counting problems for $3$-graphs. Let $E_1$ be the $3$-graph with $4$ vertices and $1$ edge. What is the number of induced $\\{K_4^3,E_1\\}$-free $3$-graphs on $n$ vertices? We show that the number of such $3$-graphs is of order $n^{\\Theta(n^2)}$. More generally, we determine asymptotically the number of induced $\\mathcal{F}$-free $3$-graphs on $n$ vertices for all families $\\mathcal{F}$ of $3$-graphs on $4$ vertices. We also provide upper bounds on the number of $r$-graphs on $n$ vertices which do not induce $i \\in L$ edges on any set of $k$ vertices, where $L \\subseteq \\big \\{0,1,\\ldots,\\binom{k}{r} \\big\\}$ is a list which does not contain $3$ consecutive integers in its complement. Our bounds are best possible up to a constant multiplicative factor in the exponent when $k = r+1$. The main tool behind our proof is counting the solutions of a constraint satisfaction problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-30T17:54:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "forbidden configurations",
      "constraint satisfaction problem",
      "forbidden structures",
      "major problems",
      "problem dates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}