{
  "id": "1608.00109",
  "version": "v1",
  "published": "2016-07-30T12:10:33.000Z",
  "updated": "2016-07-30T12:10:33.000Z",
  "title": "Monochromatic Solutions to Systems of Exponential Equations",
  "authors": [
    "Julian Sahasrabudhe"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $n\\in \\mathbb{N}$, $R$ be a binary relation on $[n]$, and $C_1(i,j),\\ldots,C_n(i,j) \\in \\mathbb{Z}$, for $i,j \\in [n]$. We define the exponential system of equations $\\mathcal{E}(R,(C_k(i,j)_{i,j,k})$ to be the system \\[ X_i^{Y_1^{C_1(i,j)} \\cdots Y_n^{C_n(i,j)} } = X_j , \\text{ for } (i,j) \\in R ,\\] in variables $X_1,\\ldots,X_n,Y_1,\\ldots,Y_n$. The aim of this paper is to classify precisely which of these systems admit a monochromatic solution ($X_i,Y_i \\not=1)$ in an arbitrary finite colouring of the natural numbers. This result could be viewed as an analogue of Rado's theorem for exponential patterns.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-30T12:10:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10"
    ],
    "keywords": [
      "monochromatic solution",
      "exponential equations",
      "systems admit",
      "binary relation",
      "exponential system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}