{
  "id": "math/0211179",
  "version": "v1",
  "published": "2002-11-11T22:25:08.000Z",
  "updated": "2002-11-11T22:25:08.000Z",
  "title": "On a hypergraph Turan problem of Frankl",
  "authors": [
    "Peter Keevash",
    "Benny Sudakov"
  ],
  "comment": "25 pages, no figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $C^{2k}_r$ be the $2k$-uniform hypergraph obtained by letting $P_1,...,P_r$ be pairwise disjoint sets of size $k$ and taking as edges all sets $P_i \\cup P_j$ with $i \\neq j$. This can be thought of as the `$k$-expansion' of the complete graph $K_r$: each vertex has been replaced with a set of size $k$. We determine the exact Turan number of $C^{2k}_3$ and the corresponding extremal hypergraph, thus confirming a conjecture of Frankl. Sidorenko has given an upper bound of $(r-2) / (r-1)$ for the Tur\\'an density of $C^{2k}_r$ for any $r$, and a construction establishing a matching lower bound when $r$ is of the form $2^p + 1$. We show that when $r = 2^p + 1$, any $C^4_r$-free hypergraph of density $(r-2)/(r-1) - o(1)$ looks approximately like Sidorenko's construction. On the other hand, when $r$ is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Tur\\'an density of $C^4_r$ to $(r-2)/(r-1) - c(r)$, where $c(r)$ is a constant depending only on $r$. The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal-Katona theorem and properties of Krawtchouck polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-11-11T22:25:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C65",
      "05D05",
      "05E35"
    ],
    "keywords": [
      "hypergraph turan problem",
      "first proving approximate structure theorems",
      "turan density",
      "upper bound",
      "exact turan number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....11179K"
    }
  }
}