{
  "id": "1401.5735",
  "version": "v1",
  "published": "2014-01-22T17:14:18.000Z",
  "updated": "2014-01-22T17:14:18.000Z",
  "title": "Universality of graphs with few triangles and anti-triangles",
  "authors": [
    "Dan Hefetz",
    "Mykhaylo Tyomkyn"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study 3-random-like graphs, that is, sequences of graphs in which the densities of triangles and anti-triangles converge to 1/8. Since the random graph ${\\mathcal G}_{n,1/2}$ is, in particular, 3-random-like, this can be viewed as a weak version of quasirandomness. We first show that 3-random-like graphs are 4-universal, that is, they contain induced copies of all 4-vertex graphs. This settles a question of Linial and Morgenstern. We then show that for larger subgraphs, 3-random-like sequences demonstrate a completely different behaviour. We prove that for every graph $H$ on $n\\geq R(10,10)$ vertices there exist 3-random-like graphs without an induced copy of $H$. Moreover, we prove that for every $\\ell$ there are 3-random-like graphs which are $\\ell$-universal but not $m$-universal when $m$ is sufficiently large compared to $\\ell$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-22T17:14:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "universality",
      "induced copy",
      "random graph",
      "anti-triangles converge",
      "weak version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.5735H"
    }
  }
}