{
  "id": "1509.02143",
  "version": "v1",
  "published": "2015-09-07T19:08:14.000Z",
  "updated": "2015-09-07T19:08:14.000Z",
  "title": "Monotone Paths in Dense Edge-Ordered Graphs",
  "authors": [
    "Kevin G. Milans"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The altitude of a graph $G$, denoted $f(G)$, is the largest integer $k$ such that under each ordering of $E(G)$, there exists a path of length $k$ which traverses edges in increasing order. In 1971, Chv\\'atal and Koml\\'os asked for $f(K_n)$, where $K_n$ is the complete graph on $n$ vertices. In 1973, Graham and Kleitman proved that $f(K_n) \\ge \\sqrt{n - 3/4} - 1/2$ and in 1984, Calderbank, Chung, and Sturtevant proved that $f(K_n) \\le (\\frac{1}{2} + o(1))n$. We show that $f(K_n) \\ge (\\frac{1}{20} - o(1))(n/\\lg n)^{2/3}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-07T19:08:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C38"
    ],
    "keywords": [
      "dense edge-ordered graphs",
      "monotone paths",
      "largest integer",
      "complete graph",
      "traverses edges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}