{
  "id": "1006.2671",
  "version": "v2",
  "published": "2010-06-14T11:12:51.000Z",
  "updated": "2013-06-13T19:35:41.000Z",
  "title": "A density version of the Halpern-Läuchli theorem",
  "authors": [
    "Pandelis Dodos",
    "Vassilis Kanellopoulos",
    "Nikolaos Karagiannis"
  ],
  "comment": "27 pages, no figures; Advances in Mathematics, to appear",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove a density version of the Halpern-L\\\"{a}uchli Theorem. This settles in the affirmative a conjecture of R. Laver. Specifically, let us say that a tree $T$ is homogeneous if $T$ has a unique root and there exists an integer $b\\meg 2$ such that every $t\\in T$ has exactly $b$ immediate successors. We show that for every $d\\meg 1$ and every tuple $(T_1,...,T_d)$ of homogeneous trees, if $D$ is a subset of the level product of $(T_1,...,T_d)$ satisfying \\[ \\limsup_{n\\to\\infty} \\frac{|D\\cap \\big(T_1(n)\\times ... \\times T_d(n)\\big)|}{|T_1(n)\\times ... \\times T_d(n)|}>0\\] then there exist strong subtrees $(S_1, ..., S_d)$ of $(T_1,...,T_d)$ having common level set such that the level product of $(S_1,...,S_d)$ is a subset of $D$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-06-13T19:35:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "density version",
      "halpern-läuchli theorem",
      "level product",
      "common level set",
      "strong subtrees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.2671D"
    }
  }
}