{
  "id": "1510.03461",
  "version": "v1",
  "published": "2015-10-12T20:59:21.000Z",
  "updated": "2015-10-12T20:59:21.000Z",
  "title": "Stability and Turán numbers of a class of hypergraphs via Lagrangians",
  "authors": [
    "Axel Brandt",
    "David Irwin",
    "Tao Jiang"
  ],
  "comment": "26 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a family of $r$-uniform hypergraphs ${\\cal F}$ (or $r$-graphs for brevity), the Tur\\'an number $ex(n,{\\cal F})$ of ${\\cal F}$ is the maximum number of edges in an $r$-graph on $n$ vertices that does not contain any member of ${\\cal F}$. A pair $\\{u,v\\}$ is covered in a hypergraph $G$ if some edge of $G$ contains $\\{u,v\\}$. Given an $r$-graph $F$ and a positive integer $p\\geq n(F)$, let $H^F_p$ denote the $r$-graph obtained as follows. Label the vertices of $F$ as $v_1,\\ldots, v_{n(F)}$. Add new vertices $v_{n(F)+1},\\ldots, v_p$. For each pair of vertices $v_i,v_j$ not covered in $F$, add a set $B_{i,j}$ of $r-2$ new vertices and the edge $\\{v_i,v_j\\}\\cup B_{i,j}$, where the $B_{i,j}$'s are pairwise disjoint over all such pairs $\\{i,j\\}$. We call $H^F_p$ the expanded $p$-clique with an embedded $F$. For a relatively large family of $F$, we show that for all sufficiently large $n$, $ex(n,H^F_p)=|T_r(n,p-1)|$, where $T_r(n,p-1)$ is the balanced complete $(p-1)$-partite $r$-graph on $n$ vertices. We also establish structural stability of near extremal graphs. Our results generalize or strengthen several earlier results and provide a class of hypergraphs for which the Tur\\'an number is exactly determined (for large $n$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-12T20:59:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "05C35",
      "05D05"
    ],
    "keywords": [
      "turán numbers",
      "turan number",
      "lagrangians",
      "uniform hypergraphs",
      "extremal graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151003461B"
    }
  }
}