{
  "id": "1505.03210",
  "version": "v1",
  "published": "2015-05-13T01:36:19.000Z",
  "updated": "2015-05-13T01:36:19.000Z",
  "title": "Turán numbers of hypergraph trees",
  "authors": [
    "Zoltán Füredi",
    "Tao Jiang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "An $r$-graph is an $r$-uniform hypergraph tree (or $r$-tree) if its edges can be ordered as $E_1,\\ldots, E_m$ such that $\\forall i>1 \\, \\exists \\alpha(i)<i$ such that $E_i\\cap (\\bigcup_{j=1}^{i-1} E_j)\\subseteq E_{\\alpha(i)}$. The Tur\\'an number $ex(n,{\\cal H})$ of an $r$-graph ${\\cal H}$ is the largest size of an $n$-vertex $r$-graph that does not contain ${\\cal H}$. A cross-cut of ${\\cal H}$ is a set of vertices in ${\\cal H}$ that contains exactly one vertex of each edge of ${\\cal H}$. The cross-cut number $\\sigma({\\cal H})$ of ${\\cal H}$ is the minimum size of a cross-cut of ${\\cal H}$. We show that for a large family of $r$-graphs (largest within a certain scope) that are embeddable in $r$-trees, $ex(n,{\\cal H})=(\\sigma-1)\\binom{n}{r-1}+o(n^{r-1})$ holds, and we establish structural stability of near extremal graphs. From stability, we establish exact results for some subfamilies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-13T01:36:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "05C65",
      "05C35"
    ],
    "keywords": [
      "turán numbers",
      "uniform hypergraph tree",
      "turan number",
      "cross-cut number",
      "extremal graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150503210F"
    }
  }
}