{
  "id": "1510.03950",
  "version": "v1",
  "published": "2015-10-14T02:33:08.000Z",
  "updated": "2015-10-14T02:33:08.000Z",
  "title": "On the Ramsey-Turán number with small $s$-independence number",
  "authors": [
    "Patrick Bennett",
    "Andrzej Dudek"
  ],
  "comment": "24 pp",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define the Ramsey-Tur\\'an number $RT_s(n,H, f)$ as the maximum number of edges in an $H$-free graph $G$ of order $n$ with $\\alpha_s(G) < f$, where $\\alpha_s(G)$ is the maximum number of vertices in a $K_s$-free induced subgraph of $G$. The Ramsey-Tur\\'an number attracted a considerable amount of attention and has been mainly studied for $f$ not too much smaller than $n$. In this paper we consider $RT_s(n,K_t, n^{\\delta})$ for fixed $\\delta<1$. We show that for an arbitrarily small $\\varepsilon>0$ and $1/2<\\delta< 1$, $RT_s(n,K_{s+1}, n^{\\delta}) = \\Omega(n^{1+\\delta-\\varepsilon})$ for all sufficiently large $s$. This is nearly optimal, since a trivial upper bound yields $RT_s(n,K_{s+1}, n^{\\delta}) = O(n^{1+\\delta})$. Furthermore, the range of $\\delta$ is as large as possible. We also consider more general cases and find bounds on $RT_s(n,K_{s+r},n^{\\delta})$ for fixed $r\\ge2$. Finally, we discuss some \"jumping\" behavior of $RT_s(n, K_{2s}, n^\\delta)$ and $RT_s(n, K_{2s+1}, n^\\delta)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-14T02:33:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "independence number",
      "ramsey-turán number",
      "maximum number",
      "ramsey-turan number",
      "trivial upper bound yields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151003950B"
    }
  }
}