{
  "id": "2404.02666",
  "version": "v1",
  "published": "2024-04-03T12:05:30.000Z",
  "updated": "2024-04-03T12:05:30.000Z",
  "title": "Nestings of BIBDs with block size four",
  "authors": [
    "Marco Buratti",
    "Donald L. Kreher",
    "Douglas R. Stinson"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the \\emph{nested point}) to every block of a $(v,k,\\lambda)$-BIBD in such a way that we end up with a partial $(v,k+1,\\lambda+1)$-BIBD. In the case where the partial $(v,k+1,\\lambda+1)$-BIBD is in fact a $(v,k+1,\\lambda+1)$-BIBD, we have a \\emph{perfect nesting}. We show that a nesting is perfect if and only if $k = 2 \\lambda + 1$. Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., $(v,3,1)$-BIBDs) when $v \\equiv 1 \\bmod 6$, as well as for some symmetric BIBDs. Here we study nestings of $(v,4,1)$-BIBDs, which are not perfect nestings. We prove that there is a nested $(v,4,1)$-BIBD if and only if $v \\equiv 1 \\text{ or } 4 \\bmod 12$, $v \\geq 13$. This is accomplished by a variety of direct and recursive constructions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-03T12:05:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B05"
    ],
    "keywords": [
      "block size",
      "perfect nestings",
      "balanced incomplete block design",
      "steiner triple systems",
      "study nestings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}