{
  "id": "2101.01757",
  "version": "v1",
  "published": "2021-01-05T19:47:12.000Z",
  "updated": "2021-01-05T19:47:12.000Z",
  "title": "A preliminary result for generalized intersecting families",
  "authors": [
    "Brian Chan"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Intersecting families and blocking sets feature prominently in extremal combinatorics. We examine the following generalization of an intersecting family investigated by Hajnal, Rothschild, and others. If $s \\geq 1$, $k \\geq 2$, and $u \\geq 1$ are integers, then say that an $s$-uniform family $\\mathcal{F}$ is $(k,u)$-intersecting if for all $A_1, A_2, \\cdots, A_k \\in \\mathcal{F}$, $|A_i \\cap A_j| \\geq u$ for some $1 \\leq i < j \\leq k$. In this note, we investigate the following parameter. If $s$, $k$, $u$, $\\ell$ are integers satisfying $s \\geq 1$, $k \\geq 2$, $1 \\leq u \\leq s$, and $2 \\leq \\ell < k$, then let $N^{(u)}_{k,\\ell}(s)$ denote the smallest integer $r$, if it exists, such that any $(k,u)$-intersecting $s$-uniform family is the union of at most $r$ families that are $(\\ell,u)$-intersecting. Using a Sunflower Lemma type argument, we prove that $N^{(u)}_{k,\\ell}(s)$ always exists and that the following inequality always holds: $$N^{(u)}_{k,\\ell}(s) \\; \\leq \\; \\bigg{\\lceil} \\dfrac{ k - 1 }{\\ell - 1} \\cdot {s \\choose u} \\bigg{\\rceil}$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-05T19:47:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "intersecting family",
      "generalized intersecting families",
      "preliminary result",
      "sunflower lemma type argument",
      "extremal combinatorics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}