{
  "id": "1605.06577",
  "version": "v1",
  "published": "2016-05-21T02:50:35.000Z",
  "updated": "2016-05-21T02:50:35.000Z",
  "title": "Avoiding patterns in matrices via a small number of changes",
  "authors": [
    "Maria Axenovich",
    "Ryan Martin"
  ],
  "comment": "6 pages",
  "journal": "SIAM J. Discrete Math., 20(1) (2006), 49--54",
  "doi": "10.1137/S0895480104445150",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let ${\\cal A}=\\{A_1,\\ldots, A_r\\}$ be a partition of a set $\\{1,\\ldots,m\\}\\times\\{1,\\ldots, n\\}$ into $r$ nonempty subsets, and $A=(a_{ij})$ be an $m\\times n$ matrix. We say that $A$ has a pattern ${\\cal A}$ provided that $a_{ij}=a_{i'j'}$ if and only if $(i,j),(i',j')\\in A_t$ for some $t\\in\\{1,\\ldots,r\\}$. In this note we study the following function $f$ defined on the set of all $m\\times n$ matrices $M$ with $s$ distinct entries: $f(M; {\\cal A})$ is the smallest number of positions where the entries of $M$ need to be changed such that the resulting matrix does not have any submatrix with pattern ${\\cal A}$. We give an asymptotically tight value for $$ f(m,n; s, {\\cal A}) = \\max\\{f(M; {\\cal A}): M \\mbox{ is an } m\\times n\\mbox{ matrix with at most } s \\mbox{ distinct entries}\\} . $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-21T02:50:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A18",
      "05C50",
      "15A99"
    ],
    "keywords": [
      "small number",
      "avoiding patterns",
      "nonempty subsets",
      "distinct entries",
      "smallest number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}