{
  "id": "2310.01923",
  "version": "v1",
  "published": "2023-10-03T09:59:27.000Z",
  "updated": "2023-10-03T09:59:27.000Z",
  "title": "Latin squares without proper subsquares",
  "authors": [
    "Jack Allsop",
    "Ian M. Wanless"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A $d$-dimensional Latin hypercube of order $n$ is a $d$-dimensional array containing symbols from a set of cardinality $n$ with the property that every axis-parallel line contains all $n$ symbols exactly once. We show that for $(n, d) \\notin \\{(4,2), (6,2)\\}$ with $d \\geq 2$ there exists a $d$-dimensional Latin hypercube of order $n$ that contains no $d$-dimensional Latin subhypercube of any order in $\\{2,\\dots,n-1\\}$. The $d=2$ case settles a 50 year old conjecture by Hilton on the existence of Latin squares without proper subsquares.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-03T09:59:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B15"
    ],
    "keywords": [
      "proper subsquares",
      "latin squares",
      "dimensional latin hypercube",
      "axis-parallel line contains",
      "dimensional array containing symbols"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}