{
  "id": "1806.02838",
  "version": "v1",
  "published": "2018-06-07T18:00:03.000Z",
  "updated": "2018-06-07T18:00:03.000Z",
  "title": "On Turán exponents of bipartite graphs",
  "authors": [
    "Tao Jiang",
    "Jie Ma",
    "Liana Yepremyan"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A long-standing conjecture of Erd\\H{o}s and Simonovits asserts that for every rational number $r\\in (1,2)$ there exists a bipartite graph $H$ such that $\\ex(n,H)=\\Theta(n^r)$. So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$, for integers $k\\geq 2$. In this paper we add a new form of rationals for which the conjecture is true; $2-2/(2k+1)$, for $k\\geq 2$. This in its turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erd\\H{o}s and Simonovits's conjecture where one replaces a single graph by a family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erd\\H{o}s and Simonovits's conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon's conjecture. We also prove the an upper bound on the Tur\\'an's number of $\\theta$-graphs in an asymmetric setting and employ this result to obtain yet another new rational exponent for Tur\\'an exponents; $r=7/5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-07T18:00:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bipartite graph",
      "turán exponents",
      "simonovitss conjecture",
      "simonovits asserts",
      "turan exponents"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}