{
  "id": "1704.07022",
  "version": "v1",
  "published": "2017-04-24T03:08:08.000Z",
  "updated": "2017-04-24T03:08:08.000Z",
  "title": "Note on the union-closed sets conjecture",
  "authors": [
    "Abigail Raz"
  ],
  "comment": "4 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The union-closed sets conjecture states that if a family of sets $\\mathcal{A} \\neq \\{\\emptyset\\}$ is union-closed, then there is an element which belongs to at least half the sets in $\\mathcal{A}$. In 2001, D. Reimer showed that the average set size of a union-closed family, $\\mathcal{A}$, is at least $\\frac{1}{2} \\log_2 |\\mathcal{A}|$. In order to do so, he showed that all union-closed families satisfy a particular condition, which in turn implies the preceding bound. Here, answering a question raised in the context of T. Gowers' polymath project on the union-closed sets conjecture, we show that Reimer's condition alone is not enough to imply that there is an element in at least half the sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-24T03:08:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "union-closed sets conjecture states",
      "polymath project",
      "turn implies",
      "average set",
      "union-closed families satisfy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}