{
  "id": "2301.01331",
  "version": "v1",
  "published": "2023-01-03T19:38:26.000Z",
  "updated": "2023-01-03T19:38:26.000Z",
  "title": "Local Configurations in Union-Closed Families",
  "authors": [
    "Jonad Pulaj",
    "Kenan Wood"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Frankl or Union-Closed Sets conjecture states that for any finite union-closed family of sets $\\mathcal{F}$ containing some nonempty set, there is some element $i$ in the ground set $U(\\mathcal F) := \\bigcup_{S \\in \\mathcal{F}} S$ of $\\mathcal{F}$ such that $i$ is in at least half of the sets in $\\mathcal{F}$. In this work, we find new values and bounds for the least integer $m$ such that any family containing $m$ distinct $k$-sets of an $n$-set $X$ satisfies Frankl's conjecture with an element of $X$. Additionally, we answer an older question of Vaughan regarding symmetry in union-closed families and we give a proof of a recent question posed by Ellis, Ivan and Leader. Finally, we introduce novel local configuration criteria to prove the conjecture for many, previously unknown classes of families.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-03T19:38:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "union-closed family",
      "novel local configuration criteria",
      "union-closed sets conjecture states",
      "satisfies frankls conjecture",
      "nonempty set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}