{
  "id": "2411.09445",
  "version": "v1",
  "published": "2024-11-14T13:48:16.000Z",
  "updated": "2024-11-14T13:48:16.000Z",
  "title": "Turán Densities for Small Hypercubes",
  "authors": [
    "David Ellis",
    "Maria-Romina Ivan",
    "Imre Leader"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "How small can a set of vertices in the $n$-dimensional hypercube $Q_n$ be if it meets every copy of $Q_d$? The asymptotic density of such a set (for $d$ fixed and $n$ large) is denoted by $\\gamma_d$. It is easy to see that $\\gamma_d \\leq 1/(d+1)$, and it is known that $\\gamma_d=1/(d+1)$ for $d \\leq 2$, but it was recently shown that $\\gamma_d < 1/(d+1)$ for $d \\geq 8$. In this paper we show that the latter phenomenon also holds for $d=7$ and $d=6$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-14T13:48:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65"
    ],
    "keywords": [
      "small hypercubes",
      "turán densities",
      "dimensional hypercube",
      "asymptotic density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}