{
  "id": "2409.12770",
  "version": "v1",
  "published": "2024-09-19T13:37:53.000Z",
  "updated": "2024-09-19T13:37:53.000Z",
  "title": "Exact Values and Bounds for Ramsey Numbers of $C_4$ Versus a Star Graph",
  "authors": [
    "Luis Boza"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The 8 unknown values of the Ramsey numbers $R(C_4,K_{1,n})$ for $n \\leq 37$ are determined, showing that $R(C_4,K_{1,27}) = 33$ and $R(C_4,K_{1,n}) = n + 7$ for $28 \\leq n \\leq 33$ or $n = 37$. Additionally, the following results are proven: $\\bullet$ If $n$ is even and $\\lceil\\sqrt{n}\\rceil$ is odd, then $R(C_4,K_{1,n}) \\leq n + \\left\\lceil\\sqrt{n-\\lceil\\sqrt{n}\\rceil+2}\\right\\rceil + 1$. $\\bullet$ If $m \\equiv 2 \\,(\\text{mod } 6)$ with $m \\geq 8$, then $R(C_4,K_{1,m^2+3}) \\leq m^2 + m + 4$. $\\bullet$ If $R(C_4,K_{1,n}) > R(C_4,K_{1,n-1})$, then $R(C_4,K_{1,2n+1-R(C_4,K_{1,n})}) \\geq n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-19T13:37:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C55",
      "G.2.2"
    ],
    "keywords": [
      "ramsey numbers",
      "exact values",
      "star graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}