{
  "id": "2310.02482",
  "version": "v1",
  "published": "2023-10-03T23:17:19.000Z",
  "updated": "2023-10-03T23:17:19.000Z",
  "title": "Strengthening the union-closed sets conjecture",
  "authors": [
    "Christopher Bouchard"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A family of sets $\\mathcal{F}$ is union-closed if $X,Y \\in \\mathcal{F} \\implies X \\cup Y \\in \\mathcal{F}$. Let $\\mathcal{A} \\neq \\{\\emptyset\\}$ be a finite union-closed family of sets. The union-closed sets conjecture, also called Frankl's conjecture, states that there exists an element of $\\bigcup_{A \\in \\mathcal{A}} A $ that appears in at least $\\frac{|\\mathcal{A}|}{2}$ members of $\\mathcal{A}$. Let $\\mathcal{A}_B=\\{A\\in\\mathcal{A}|A \\cap B = B\\}$ and $\\mathcal{A}_{\\bar{B}}=\\{A\\in\\mathcal{A}|A \\cap B = \\emptyset\\}$ where $B \\subseteq \\bigcup_{A \\in \\mathcal{A}}A$. Further, let ${S \\choose k}$ be the set of all $k$-element subsets of a set $S$, and $[n]=\\{1,2,\\cdots,n\\}=\\bigcup_{A \\in \\mathcal{A}}A$. The union-closed sets conjecture can then be stated as $\\exists B \\in {[n] \\choose 1}$ $|\\mathcal{A}_{B}| \\geq |\\mathcal{A}_{\\bar{B}}|$. With this notation, we introduce the stronger conjecture that $\\forall x \\in [n]$ $\\exists B \\in {[n] \\choose n+1-x}$ $|\\mathcal{A}_B| \\geq |\\mathcal{A}_{\\bar{B}}|$, and we prove the new conjecture for $x \\in [\\lceil \\frac{n}{3} \\rceil + 1]$, where $\\lceil \\frac{n}{3} \\rceil$ is the smallest integer greater than or equal to $\\frac{n}{3}$. Other related conjectures are investigated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-03T23:17:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "union-closed sets conjecture",
      "smallest integer greater",
      "element subsets",
      "frankls conjecture",
      "stronger conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}