{
  "id": "2108.11201",
  "version": "v1",
  "published": "2021-08-25T12:30:23.000Z",
  "updated": "2021-08-25T12:30:23.000Z",
  "title": "Ramsey numbers of quadrilateral versus books",
  "authors": [
    "Tianyu Li",
    "Qizhong Lin",
    "Xing Peng"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A book $B_n$ is a graph which consists of $n$ triangles sharing a common edge. In this paper, we study Ramsey numbers of quadrilateral versus books. Previous results give the exact value of $r(C_4,B_n)$ for $1\\le n\\le 14$. We aim to show the exact value of $r(C_4,B_n)$ for infinitely many $n$. To achieve this, we first prove that $r(C_4,B_{(m-1)^2+(t-2)})\\le m^2+t$ for $m\\ge4$ and $0 \\leq t \\leq m-1$. This improves upon a result by Faudree, Rousseau and Sheehan (1978) which states that \\begin{align*} r(C_4,B_n)\\le g(g(n)), \\;\\;\\text{where}\\;\\;g(n)=n+\\lfloor\\sqrt{n-1}\\rfloor+2. \\end{align*} Combining the new upper bound and constructions of $C_4$-free graphs, we are able to determine the exact value of $r(C_4,B_n)$ for infinitely many $n$. As a special case, we show $r(C_4,B_{q^2-q-2}) = q^2+q-1$ for all prime power $q\\ge4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-25T12:30:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exact value",
      "quadrilateral",
      "study ramsey numbers",
      "common edge",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}