{
  "id": "2004.01175",
  "version": "v1",
  "published": "2020-04-02T17:55:52.000Z",
  "updated": "2020-04-02T17:55:52.000Z",
  "title": "On the Clique Number of Paley Graphs of Prime Power Order",
  "authors": [
    "Chi Hoi Yip"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Finding a reasonably good upper bound for the clique number of Paley graph is an old and open problem in additive combinatorics. A recent breakthrough by Hanson and Petridis using Stepanov's method gives an improved upper bound on $\\mathbb{F}_p$, where $p \\equiv 1 \\pmod 4$. We extend their idea to the finite field $\\mathbb{F}_q$, where $q=p^{2s+1}$ for a prime $p\\equiv 1 \\pmod 4$ and a non-negative integer $s$. We show the clique number of the Paley graph over $\\mathbb{F}_{p^{2s+1}}$ is at most $\\min \\bigg(p^s \\bigg\\lceil \\sqrt{\\frac{p}{2}} \\bigg\\rceil, \\sqrt{\\frac{q}{2}} + \\frac{p^s+1}{4} + \\frac{\\sqrt{2p}}{32} p^{s-1}\\bigg)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-02T17:55:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06"
    ],
    "keywords": [
      "clique number",
      "paley graph",
      "prime power order",
      "upper bound",
      "open problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}