{
  "id": "2412.12466",
  "version": "v1",
  "published": "2024-12-17T01:58:57.000Z",
  "updated": "2024-12-17T01:58:57.000Z",
  "title": "Latin Squares whose transversals share many entries",
  "authors": [
    "Afsane Ghafari",
    "Ian M. Wanless"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that, for all even $n\\geq10$, there exists a latin square of order $n$ with at least one transversal, yet all transversals coincide on $ \\big\\lfloor n/6 \\big\\rfloor$ entries. These latin squares have at least $ 19 n^2/36 + O(n)$ transversal-free entries. We also prove that for all odd $m\\geq 3$, there exists a latin square of order $n=3m$ divided into nine $m\\times m$ subsquares, where every transversal hits each of these subsquares at least once.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-17T01:58:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B15"
    ],
    "keywords": [
      "latin square",
      "transversals share",
      "transversals coincide",
      "transversal hits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}