{
  "id": "2404.04867",
  "version": "v1",
  "published": "2024-04-07T08:24:15.000Z",
  "updated": "2024-04-07T08:24:15.000Z",
  "title": "A Bollobás-type problem: from root systems to Erdős-Ko-Rado",
  "authors": [
    "Patrick J. Browne",
    "Qëndrim R. Gashi",
    "Padraig Ó Catháin"
  ],
  "comment": "7 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Motivated by an Erd\\H{o}s--Ko--Rado type problem on sets of strongly orthogonal roots in the $A_{\\ell}$ root system, we estimate bounds for the size of a family of pairs $(A_{i}, B_{i})$ of $k$-subsets in $\\{ 1, 2, \\ldots, n\\}$ such that $A_{i} \\cap B_{j}= \\emptyset$ and $|A_{i} \\cap A_{j}| + |B_{i} \\cap B_{j}| = k$ for all $i \\neq j$. This is reminiscent of a classic problem of Bollob\\'as. We provide upper and lower bounds for this problem, relying on classical results of extremal combinatorics and an explicit construction using the incidence matrix of a finite projective plane.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-07T08:24:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "17B22"
    ],
    "keywords": [
      "root system",
      "bollobás-type problem",
      "erdős-ko-rado",
      "finite projective plane",
      "strongly orthogonal roots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}