{
  "id": "1410.4900",
  "version": "v1",
  "published": "2014-10-18T02:34:41.000Z",
  "updated": "2014-10-18T02:34:41.000Z",
  "title": "Sets of natural numbers with proscribed subsets",
  "authors": [
    "Kevin O'Bryant"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Fix $A$, a family of subsets of natural numbers, and let $G_A(n)$ be the maximum cardinality of a subset of $\\{1,2,..., n\\}$ that does not have any subset in $A$. We consider the general problem of giving upper bounds on $G_A(n)$ and give some new upper bounds on some families that are closed under dilation. Specific examples include sets that do not contain any geometric progression of length $k$ with integer ratio, sets that do not contain any geometric progression of length $k$ with rational ratio, and sets of integers that do not contain multiplicative squares, i.e., nontrivial sets of the form $\\{a, ar, as, ars\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-18T02:34:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B05",
      "11B25",
      "11B75",
      "11B83",
      "05D10"
    ],
    "keywords": [
      "natural numbers",
      "proscribed subsets",
      "geometric progression",
      "maximum cardinality",
      "specific examples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.4900O"
    }
  }
}