{
  "id": "2109.07539",
  "version": "v1",
  "published": "2021-09-15T19:11:50.000Z",
  "updated": "2021-09-15T19:11:50.000Z",
  "title": "Erdős-Ginzburg-Ziv type generalizations for linear equations and linear inequalities in three variables",
  "authors": [
    "Mario Huicochea",
    "Amanda Montejano"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For any linear inequality in three variables $\\mathcal{L}$, we determine (if it exist) the smallest integer $R(\\mathcal{L}, \\mathbb{Z}/3\\mathbb{Z})$ such that: for every mapping $\\chi :[1,n] \\to \\{0,1,2\\}$, with $n\\geq R(\\mathcal{L}, \\mathbb{Z}/3\\mathbb{Z})$, there is a solution $(x_1,x_2,x_3)\\in [1,n]^3$ of $\\mathcal{L}$ with $\\chi(x_1)+\\chi(x_2)+\\chi(x_3)\\equiv 0$ (mod $3$). Moreover, we prove that $R(\\mathcal{L}, \\mathbb{Z}/3\\mathbb{Z})=R(\\mathcal{L}, 2)$, where $R(\\mathcal{L}, 2)$ denotes the classical $2$-color Rado number, that is, the smallest integer (provided it exist) such that for every $2$-coloring of $[1,n]$, with $n\\geq R(\\mathcal{L}, 2)$, there exist a monochromatic solution of $\\mathcal{L}$. Thus, we get an Erd\\H{o}s-Ginzburg-Ziv type generalization for all lineal inequalities in three variables having a solution in the positive integers. We also show a number of families of linear equations in three variables $\\mathcal{L}$ such that they do not admit such Erd\\H{o}s-Ginzburg-Ziv type generalization, named $R(\\mathcal{L}, \\mathbb{Z}/3\\mathbb{Z})\\neq R(\\mathcal{L}, 2)$. At the end of this paper some questions are proposed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-15T19:11:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "erdős-ginzburg-ziv type generalizations",
      "linear equations",
      "linear inequality",
      "smallest integer",
      "color rado number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}