{
  "id": "1904.00146",
  "version": "v1",
  "published": "2019-03-30T04:22:36.000Z",
  "updated": "2019-03-30T04:22:36.000Z",
  "title": "An Improved Error Term for Tur$\\acute{\\rm a}$n Number of Expanded Non-degenerate 2-graphs",
  "authors": [
    "Yucong Tang",
    "Xin Xu",
    "Guiying Yan"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For a 2-graph $F$, let $H_F^{(r)}$ be the $r$-graph obtained from $F$ by enlarging each edge with a new set of $r-2$ vertices. We show that if $\\chi(F)=\\ell>r \\geq 2$, then $ {\\rm ex}(n,H_F^{(r)})= t_r (n,\\ell-1)+ \\Theta( {\\rm biex}(n,F)n^{r-2}),$ where $t_r (n,\\ell-1)$ is the number of edges of an $n$-vertex complete balanced $\\ell-1$ partite $r$-graph and ${\\rm biex}(n,F)$ is the extremal number of the decomposition family of $F$. Since ${\\rm biex}(n,F)=O(n^{2-\\gamma})$ for some $\\gamma>0$, this improves on the bound ${\\rm ex}(n,H_F^{(r)})= t_r (n,\\ell-1)+ o(n^r)$ by Mubayi (2016). Furthermore, our result implies that ${\\rm ex}(n,H_F^{(r)})= t_r (n,\\ell-1)$ when $F$ is edge-critical, which is an extension of the result of Pikhurko (2013).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-30T04:22:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "05C35"
    ],
    "keywords": [
      "error term",
      "expanded non-degenerate",
      "result implies",
      "vertex complete",
      "extremal number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}