{
  "id": "2010.01784",
  "version": "v1",
  "published": "2020-10-05T05:12:10.000Z",
  "updated": "2020-10-05T05:12:10.000Z",
  "title": "On the directions determined by Cartesian products and the clique number of generalized Paley graphs",
  "authors": [
    "Chi Hoi Yip"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "It is known that the number of directions formed by a Cartesian product $A \\times B \\subset AG(2,p)$ is at least $|A||B| - \\min\\{|A|,|B|\\} + 2$, provided $p$ is prime and $|A||B|<p$. This implies the best known upper bound on the clique number of the Paley graph over $\\mathbb{F}_p$. In this paper, we extend this result to $AG(2,q)$, where $q$ is a prime power. We also give improved upper bounds on the clique number of generalized Paley graphs over $\\mathbb{F}_q$. In particular, for a cubic Paley graph, we improve the trivial bound $\\sqrt{q}$ to $0.769\\sqrt{q}+1$. In general, as an application of our key result on the number of directions, for any positive function $h$ such that $h(x)=o(x)$ as $x \\to \\infty$, we improve the bound to $\\sqrt{q}-h(p)$ for almost all non-squares $q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-05T05:12:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06"
    ],
    "keywords": [
      "generalized paley graphs",
      "clique number",
      "cartesian product",
      "directions",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}