{
  "id": "1909.11602",
  "version": "v1",
  "published": "2019-09-23T18:08:17.000Z",
  "updated": "2019-09-23T18:08:17.000Z",
  "title": "Design Theory and some Forbidden Configurations",
  "authors": [
    "R. P. Anstee",
    "Farzin Barekat",
    "Zachary Pellegrin"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1909.07580",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper we relate t-designs to a forbidden configuration problem in extremal set theory. Let 1_t 0_l denote a column of t 1's on top of l 0's. We assume t>l. Let q. (1_t 0_l) denote the (t+l)xq matrix consisting of t rows of q 1's and l rows of q 0's. We consider extremal problems for matrices avoiding certain submatrices. Let A be a (0,1)-matrix forbidding any (t+l)x(\\lambda+2) submatrix (\\lambda+2). (1_t 0_l) . Assume A is m-rowed and only columns of sum t+1,t+2,... ,m-l are allowed to be repeated. Assume that A has the maximum number of columns subject to the given restrictions. Assume m is sufficiently large. Then A has each column of sum 0,1,... ,t and m-l+1,m-l+2,..., m exactly once and, given the appropriate divisibility condition, the columns of sum t+1 correspond to a t-design with block size t+1 and parameter \\lambda and there are no other columns. The proof derives a basic upper bound on the number of columns of A by a pigeonhole argument and then a careful argument, for large m, reduces the bound by a substantial amount down to the value given by design based constructions. We extend in a few directions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-23T18:08:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "design theory",
      "extremal set theory",
      "forbidden configuration problem",
      "appropriate divisibility condition",
      "basic upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}