{
  "id": "2211.02338",
  "version": "v1",
  "published": "2022-11-04T09:31:08.000Z",
  "updated": "2022-11-04T09:31:08.000Z",
  "title": "Some exact values on Ramsey numbers related to fans",
  "authors": [
    "Qinghong Zhao",
    "Bing Wei"
  ],
  "comment": "10 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "For two given graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the smallest integer $N$ such that any red-blue edge-coloring of the complete graph $K_N$ contains a red $F$ or a blue $H$. When $F=H$, we simply write $R_2(H)$. For an positive integer $n$, let $K_{1,n}$ be a star with $n+1$ vertices, $F_n$ be a fan with $2n+1$ vertices consisting of $n$ triangles sharing one common vertex, and $nK_3$ be a graph with $3n$ vertices obtained from the disjoint union of $n$ triangles. In 1975, Burr, Erd\\H{o}s and Spencer \\cite{B} proved that $R_2(nK_3)=5n$ for $n\\ge2$. However, determining the exact value of $R_2(F_n)$ is notoriously difficult. So far, only $R_2(F_2)=9$ has been proved. Notice that both $F_n$ and $nK_3$ contain $n$ triangles and $|V(F_n)|<|V(nK_3)|$ for all $n\\ge 2$. Chen, Yu and Zhao (2021) speculated that $R_2(F_n)\\le R_2(nK_3)=5n$ for $n$ sufficiently large. In this paper, we first prove that $R(K_{1,n},F_n)=3n-\\varepsilon$ for $n\\ge1$, where $\\varepsilon=0$ if $n$ is odd and $\\varepsilon=1$ if $n$ is even. Applying the exact values of $R(K_{1,n},F_n)$, we will confirm $R_2(F_n)\\le 5n$ for $n=3$ by showing that $R_2(F_3)=14$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-04T09:31:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exact value",
      "ramsey number",
      "common vertex",
      "disjoint union",
      "complete graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}