{
  "id": "2002.06795",
  "version": "v1",
  "published": "2020-02-17T06:30:54.000Z",
  "updated": "2020-02-17T06:30:54.000Z",
  "title": "On the Turán number of 1-subdivision of $K_{3,t}$",
  "authors": [
    "Tao Zhang",
    "Zixiang Xu",
    "Gennian Ge"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a graph $H$, the 1-subdivision of $H$, denoted by $H'$, is the graph obtained by replacing the edges of $H$ by internally disjoint paths of length 2. Recently, Conlon, Janzer and Lee (arXiv: 1903.10631) asked the following question: For any integer $s\\ge2$, estimate the smallest $t$ such that $\\textup{ex}(n,K_{s,t}')=\\Omega(n^{\\frac{3}{2}-\\frac{1}{2s}})$. In this paper, we consider the case $s=3$. More precisely, we provide an explicit construction giving \\begin{align*} \\text{ex}(n,K_{3,30}')=\\Omega(n^{\\frac{4}{3}}), \\end{align*} which reduces the estimation for the smallest value of $t$ from a magnitude of $10^{56}$ to the number $30$. The construction is algebraic, which is based on some equations over finite fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-17T06:30:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "turán number",
      "finite fields",
      "smallest value",
      "internally disjoint paths",
      "explicit construction giving"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}