{
  "id": "2405.12820",
  "version": "v1",
  "published": "2024-05-21T14:25:58.000Z",
  "updated": "2024-05-21T14:25:58.000Z",
  "title": "Weak and Strong Nestings of BIBDs",
  "authors": [
    "Douglas R. Stinson"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We study two types of nestings of balanced incomplete block designs (BIBDs). In both types of nesting, we wish to add a point (the nested point) to every block of a $(v,k,\\lambda)$-BIBD in such a way that we end up with a partial $(w,k+1,\\lambda+1)$-BIBD for some $w \\geq v$. In the case where $w > v$, we are introducing $w-v$ new points. This is called a weak nesting. A strong nesting satisfies the stronger property that no pair containing a new point occurs more than once in the partial $(w,k+1,\\lambda+1)$-BIBD. In both cases, the goal is to minimize $w$. We prove lower bounds on $w$ as a function of $v$, $k$ and $\\lambda$ and we find infinite classes of $(v,2,1)$- and $(v,3,2)$-BIBDs that have optimal nestings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-21T14:25:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B05"
    ],
    "keywords": [
      "balanced incomplete block designs",
      "infinite classes",
      "stronger property",
      "strong nesting satisfies",
      "lower bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}