{
  "id": "1908.06480",
  "version": "v1",
  "published": "2019-08-18T17:08:23.000Z",
  "updated": "2019-08-18T17:08:23.000Z",
  "title": "On the local structure of oriented graphs -- a case study in flag algebras",
  "authors": [
    "Shoni Gilboa",
    "Roman Glebov",
    "Dan Hefetz",
    "Nati Linial",
    "Avraham Morgenstern"
  ],
  "comment": "44 pages, 11 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be an $n$-vertex oriented graph. Let $t(G)$ (respectively $i(G)$) be the probability that a random set of $3$ vertices of $G$ spans a transitive triangle (respectively an independent set). We prove that $t(G) + i(G) \\geq \\frac{1}{9}-o_n(1)$. Our proof uses the method of flag algebras that we supplement with several steps that make it more easily comprehensible. We also prove a stability result and an exact result. Namely, we describe an extremal construction, prove that it is essentially unique, and prove that if $H$ is sufficiently far from that construction, then $t(H) + i(H)$ is significantly larger than $\\frac{1}{9}$. We go to greater technical detail than is usually done in papers that rely on flag algebras. Our hope is that as a result this text can serve others as a useful introduction to this powerful and beautiful method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-18T17:08:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "flag algebras",
      "case study",
      "local structure",
      "independent set",
      "random set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}