{
  "id": "1707.03091",
  "version": "v1",
  "published": "2017-07-11T00:46:23.000Z",
  "updated": "2017-07-11T00:46:23.000Z",
  "title": "Supersaturation of Even Linear Cycles in Linear Hypergraphs",
  "authors": [
    "Tao Jiang",
    "Liana Yepremyan"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A classic result of Erd\\H{o}s and, independently, of Bondy and Simonovits says that the maximum number of edges in an $n$-vertex graph not containing $C_{2k}$, the cycle of length $2k$, is $O( n^{1+1/k})$. Simonovits established a corresponding supersaturation result for $C_{2k}$'s, showing that there exist positive constants $C,c$ depending only on $k$ such that every $n$-vertex graph $G$ with $e(G)\\geq Cn^{1+1/k}$ contains at least $c\\left(\\frac{e(G)}{v(G)}\\right)^{2k}$ many copies of $C_{2k}$, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper, we extend Simonovits' result to a supersaturation result of $r$-uniform linear cycles of even length in $r$-uniform linear hypergraphs. Our proof is self-contained and includes the $r=2$ case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-11T00:46:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vertex graph",
      "supersaturation result",
      "uniform linear cycles",
      "uniform linear hypergraphs",
      "general host graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}