{
  "id": "2205.10789",
  "version": "v1",
  "published": "2022-05-22T10:05:40.000Z",
  "updated": "2022-05-22T10:05:40.000Z",
  "title": "Some intersection theorems for finite sets",
  "authors": [
    "Mengyu Cao",
    "Mei Lu",
    "Benjian Lv",
    "Kaishun Wang"
  ],
  "comment": "28 pages. arXiv admin note: text overlap with arXiv:2201.06339",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $n$, $r$, $k_1,\\ldots,k_r$ and $t$ be positive integers with $r\\geq 2$, and $\\mathcal{F}_i\\ (1\\leq i\\leq r)$ a family of $k_i$-subsets of an $n$-set $V$. The families $\\mathcal{F}_1,\\ \\mathcal{F}_2,\\ldots,\\mathcal{F}_r$ are said to be $r$-cross $t$-intersecting if $|F_1\\cap F_2\\cap\\cdots\\cap F_r|\\geq t$ for all $F_i\\in\\mathcal{F}_i\\ (1\\leq i\\leq r),$ and said to be non-trivial if $|\\cap_{1\\leq i\\leq r}\\cap_{F\\in\\mathcal{F}_i}F|<t$. If the $r$-cross $t$-intersecting families $\\mathcal{F}_1,\\ldots,\\mathcal{F}_r$ satisfy $\\mathcal{F}_1=\\cdots=\\mathcal{F}_r=\\mathcal{F}$, then $\\mathcal{F}$ is well known as $r$-wise $t$-intersecting family. In this paper, we describe the structure of non-trivial $r$-wise $t$-intersecting families with maximum size, and give a stability result for these families. We also determine the structure of non-trivial $2$-cross $t$-intersecting families with maximum product of their sizes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-22T10:05:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "05A10"
    ],
    "keywords": [
      "finite sets",
      "intersection theorems",
      "intersecting family",
      "non-trivial",
      "stability result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}