{
  "id": "2111.08076",
  "version": "v2",
  "published": "2021-11-15T20:53:02.000Z",
  "updated": "2023-08-03T19:53:10.000Z",
  "title": "Sidon-Ramsey and $B_{h}$-Ramsey numbers",
  "authors": [
    "Manuel A. Espinosa-García",
    "Amanda Montejano",
    "Edgardo Roldán-Pensado",
    "J. David Suárez"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a given positive integer $k$, the Sidon-Ramsey number $\\SR(k)$ is defined as the minimum value of $n$ such that, in every partition of the set $[1, n]$ into $k$ parts, there exists a part that contains two distinct pairs of numbers with the same sum. In other words, there is a part that is not a Sidon set. In this paper, we investigate the asymptotic behavior of this parameter and two generalizations of it. The first generalization involves replacing pairs of numbers with $h$-tuples, such that in every partition of $[1, n]$ into $k$ parts, there exists a part that contains two distinct $h$-tuples with the same sum. Alternatively, there is a part that is not a $B_h$ set. The second generalization considers the scenario where the interval $[1, n]$ is substituted with a non-necessarily symmetric $d$-dimensional box of the form $\\prod_{i=1}^d[1,n_i]$. For the general case of $h\\geq 3$ and non-symmetric boxes, before applying our method to obtain the Ramsey-type result, we needed to establish an upper bound for the corresponding density parameter.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-08-03T19:53:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ramsey numbers",
      "second generalization",
      "corresponding density parameter",
      "sidon-ramsey number",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}