{
  "id": "1512.00083",
  "version": "v1",
  "published": "2015-11-30T23:06:05.000Z",
  "updated": "2015-11-30T23:06:05.000Z",
  "title": "New Conjectures for Union-Closed Families",
  "authors": [
    "Jonad Pulaj",
    "Annie Raymond",
    "Dirk Theis"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO",
    "math.OC"
  ],
  "abstract": "The Frankl conjecture, also known as the union-closed sets conjecture, states that there exists an element in at least half of the sets of any (non-empty) union-closed family. From an optimization point of view, one could instead prove that $2a$ is an upperbound to the number of sets in a union-closed family with $n$ elements where each element is in at most $a$ sets for all $a,n\\in \\mathbb{N}^+$. Similarly, one could prove that the minimum number of sets containing the most frequent element in a (non-empty) union-closed family with $m$ sets and $n$ elements is at least $\\frac{m}{2}$ for any $m,n\\in \\mathbb{N}^+$. Formulating these problems as integer programs we observe that computed optimal values do not vary with $n$. We formalize these observations as conjectures, and show that they are not equivalent to the Frankl conjecture while still having wide-reaching implications if proven true. Finally, we partially prove the new conjectures and discuss possible approaches to solve them completely.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-30T23:06:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "union-closed family",
      "frankl conjecture",
      "optimization point",
      "union-closed sets conjecture",
      "minimum number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}