{
  "id": "1406.0107",
  "version": "v1",
  "published": "2014-05-31T20:09:34.000Z",
  "updated": "2014-05-31T20:09:34.000Z",
  "title": "Long paths in the distance graph over large subsets of vector spaces over finite fields",
  "authors": [
    "M. Bennett",
    "J. Chapman",
    "D. Covert",
    "D. Hart",
    "A. Iosevich",
    "J. Pakianathan"
  ],
  "categories": [
    "math.CO",
    "math.CA",
    "math.NT"
  ],
  "abstract": "Let $E \\subset {\\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices corresponding to vectors $x,y \\in E$ by an edge if $||x-y||={(x_1-y_1)}^2+\\dots+{(x_d-y_d)}^2=1$. We shall prove that if the size of $E$ is sufficiently large, then the distance graph of $E$ contains long non-overlapping paths and vertices of high degree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-31T20:09:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C10"
    ],
    "keywords": [
      "distance graph",
      "finite field",
      "large subsets",
      "long paths",
      "dimensional vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.0107B"
    }
  }
}