{
  "id": "2211.02540",
  "version": "v1",
  "published": "2022-11-04T16:03:21.000Z",
  "updated": "2022-11-04T16:03:21.000Z",
  "title": "On hierarchically closed fractional intersecting families",
  "authors": [
    "Niranjan Balachandran",
    "Srimanta Bhattacharya",
    "Krishn Vishwas Kher",
    "Rogers Mathew",
    "Brahadeesh Sankarnarayanan"
  ],
  "comment": "18 pages, 0 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a set $L$ of positive proper fractions and a positive integer $r \\geq 2$, a fractional $r$-closed $L$-intersecting family is a collection $\\mathcal{F} \\subset \\mathcal{P}([n])$ with the property that for any $2 \\leq t \\leq r$ and $A_1, \\dotsc, A_t \\in \\mathcal{F}$ there exists $\\theta \\in L$ such that $\\lvert A_1 \\cap \\dotsb \\cap A_t \\rvert \\in \\{ \\theta \\lvert A_1 \\rvert, \\dotsc, \\theta \\lvert A_t \\rvert\\}$. In this paper we show that for $r \\geq 3$ and $L = \\{\\theta\\}$ any fractional $r$-closed $\\theta$-intersecting family has size at most linear in $n$, and this is best possible up to a constant factor. We also show that in the case $\\theta = 1/2$ we have a tight upper bound of $\\lfloor \\frac{3n}{2} \\rfloor - 2$ and that a maximal $r$-closed $(1/2)$-intersecting family is determined uniquely up to isomorphism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-04T16:03:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "05B99",
      "03E05"
    ],
    "keywords": [
      "intersecting family",
      "hierarchically closed fractional intersecting families",
      "tight upper bound",
      "positive proper fractions",
      "constant factor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}