{
  "id": "1409.6921",
  "version": "v1",
  "published": "2014-09-24T12:17:55.000Z",
  "updated": "2014-09-24T12:17:55.000Z",
  "title": "Improved algorithms for colorings of simple hypergraphs and applications",
  "authors": [
    "Jakub Kozik",
    "Dmitry Shabanov"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "The paper deals with extremal problems concerning colorings of hypergraphs. By using a random recoloring algorithm we show that any $n$-uniform simple (i.e. every two distinct edges share at most one vertex) hypergraph $H$ with maximum edge degree at most \\[ \\Delta(H)\\leq c\\cdot nr^{n-1}, \\] is $r$-colorable, where $c>0$ is an absolute constant. %We prove also that similar result holds for $b$-simple hypergraphs. As an application of our proof technique we establish a new lower bound for Van der Waerden number $W(n,r)$, the minimum $N$ such that in any $r$-coloring of the set $\\{1,...,N\\}$ there exists a monochromatic arithmetic progression of length $n$. We show that \\[ W(n,r)>c\\cdot r^{n-1}, \\] for some absolute constant $c>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-24T12:17:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C65",
      "05D40",
      "G.2.1",
      "G.2.2",
      "G.3"
    ],
    "keywords": [
      "simple hypergraphs",
      "application",
      "van der waerden number",
      "absolute constant",
      "distinct edges share"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.6921K"
    }
  }
}