{
  "id": "2405.17639",
  "version": "v1",
  "published": "2024-05-27T20:20:29.000Z",
  "updated": "2024-05-27T20:20:29.000Z",
  "title": "Some new Bollobás-type inequalities",
  "authors": [
    "Erfei Yue"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A family of disjoint pairs of finite sets $\\mathcal{P}=\\{(A_i,B_i)\\mid i\\in[m]\\}$ is called a Bollob\\'as system if $A_i\\cap B_j\\neq\\emptyset$ for every $i\\neq j$, and a skew Bollob\\'as system if $A_i\\cap B_j\\neq\\emptyset$ for every $i<j$. Bollob\\'as proved that for a Bollob\\'as system, the inequality \\begin{equation*} \\sum_{i=1}^m\\binom{|A_i|+|B_i|}{|A_i|}^{-1}\\leq 1 \\end{equation*} holds. Heged\\\"{u}s and Frankl generalized this theorem to skew Bollob\\'as systems with the inequality \\begin{equation*} \\sum_{i=1}^m\\binom{|A_i|+|B_i|}{|A_i|}^{-1}\\leq 1+n, \\end{equation*} provided $A_i,B_i\\subseteq [n]$. In this paper, we improve this inequality to \\begin{equation*} \\sum_{i=1}^m \\left((1+|A_i|+|B_i|) \\binom{|A_i|+|B_i|}{|A_i|}\\right)^{-1} \\leq 1 \\end{equation*} with probabilistic method. We also generalize this result to partitions of sets on both symmetric and skew cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-27T20:20:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "inequality",
      "bollobás-type inequalities",
      "skew bollobas system",
      "disjoint pairs",
      "finite sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}