{
  "id": "1902.07819",
  "version": "v1",
  "published": "2019-02-21T00:10:37.000Z",
  "updated": "2019-02-21T00:10:37.000Z",
  "title": "On the existence of dense substructures in finite groups",
  "authors": [
    "Ching Wong"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Fix $k \\geq 6$. We prove that any large enough finite group $G$ contains $k$ elements which span quadratically many triples of the form $(a,b,ab) \\in S \\times G$, given any dense set $S \\subseteq G \\times G$. The quadratic bound is asymptotically optimal. In particular, this provides an elementary proof of a special case of a conjecture of Brown, Erd\\H{o}s and S\\'{o}s. We remark that the result was recently discovered independently by Nenadov, Sudakov and Tyomkyn.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-21T00:10:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite group",
      "dense substructures",
      "quadratic bound",
      "special case",
      "elementary proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}