{
  "id": "1211.5771",
  "version": "v2",
  "published": "2012-11-25T16:09:32.000Z",
  "updated": "2012-11-29T14:33:00.000Z",
  "title": "Capturing Forms in Dense Subsets of Finite Fields",
  "authors": [
    "Brandon Hanson"
  ],
  "comment": "Corrected typos. Added reference to other work on the subject",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "An open problem of arithmetic Ramsey theory asks if given a finite $r$-colouring $c:\\mathbb{N}\\to\\{1,...,r\\}$ of the natural numbers, there exist $x,y\\in \\mathbb{N}$ such that $c(xy)=c(x+y)$ apart from the trivial solution $x=y=2$. More generally, one could replace $x+y$ with a binary linear form and $xy$ with a binary quadratic form. In this paper we examine the analogous problem in a finite field $\\mathbb{F}_q$. Specifically, given a linear form $L$ and a quadratic from $Q$ in two variables, we provide estimates on the necessary size of $A\\subset \\mathbb{F}_q$ to guarantee that $L(x,y)$ and $Q(x,y)$ are elements of $A$ for some $x,y\\in\\mathbb{F}_q$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-11-29T14:33:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T24",
      "05D10"
    ],
    "keywords": [
      "finite field",
      "dense subsets",
      "capturing forms",
      "arithmetic ramsey theory asks",
      "binary linear form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.5771H"
    }
  }
}