{
  "id": "2410.21651",
  "version": "v1",
  "published": "2024-10-29T01:41:24.000Z",
  "updated": "2024-10-29T01:41:24.000Z",
  "title": "Optimization Tools for Computing Colorings of $[1,\\cdots ,n]$ with Few Monochromatic Solutions on $3$-variable Linear Equations",
  "authors": [
    "Jesús A. De Loera",
    "Denae Ventura",
    "Liuyue Wang",
    "William J. Wesley"
  ],
  "comment": "23 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations $E$ there is a threshold value $R_k(E)$ (the Rado number of $E$) such that for any $k$-coloring of the integers in the interval $[1,n]$, with $n \\ge R_k(E)$, there exists at least one monochromatic solution. But one can further ask, how many monochromatic solutions is the minimum possible in terms of $n$? Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-29T01:41:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "monochromatic solution",
      "variable linear equations",
      "optimization tools",
      "computing colorings",
      "arithmetic ramsey theory says"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}