{
  "id": "2201.03663",
  "version": "v2",
  "published": "2022-01-10T21:49:44.000Z",
  "updated": "2022-12-01T14:04:20.000Z",
  "title": "Chain-dependent Conditions in Extremal Set Theory",
  "authors": [
    "Dániel T. Nagy",
    "Kartal Nagy"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In extremal set theory our usual goal is to find the maximal size of a family of subsets of an $n$-element set satisfying a condition. A condition is called chain-dependent, if it is satisfied for a family if and only if it is satisfied for its intersections with the $n!$ full chains. We introduce a method to handle problems with such conditions, then show how it can be used to prove three classic theorems. Then, a theorem about families containing no two sets such that $A\\subset B$ and $\\lambda \\cdot |A| \\le |B|$ is proved. Finally, we investigate problems where instead of the size of the family, the number of $\\ell$-chains is maximized.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-12-01T14:04:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "extremal set theory",
      "chain-dependent conditions",
      "usual goal",
      "classic theorems",
      "handle problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}