{
  "id": "1209.4988",
  "version": "v2",
  "published": "2012-09-22T11:41:20.000Z",
  "updated": "2013-06-13T19:33:36.000Z",
  "title": "Measurable events indexed by products of trees",
  "authors": [
    "Pandelis Dodos",
    "Vassilis Kanellopoulos",
    "Konstantinos Tyros"
  ],
  "comment": "37 pages, no figures; Combinatorica, to appear. This article is a sequel to and draws heavily from arXiv:1105.2419",
  "categories": [
    "math.CO"
  ],
  "abstract": "A tree $T$ is said to be homogeneous if it is uniquely rooted and there exists an integer $b\\meg 2$, called the branching number of $T$, such that every $t\\in T$ has exactly $b$ immediate successors. A vector homogeneous tree $\\mathbf{T}$ is a finite sequence $(T_1,...,T_d)$ of homogeneous trees and its level product $\\otimes\\mathbf{T}$ is the subset of the cartesian product $T_1\\times ...\\times T_d$ consisting of all finite sequences $(t_1,...,t_d)$ of nodes having common length. We study the behavior of measurable events in probability spaces indexed by the level product $\\otimes\\mathbf{T}$ of a vector homogeneous tree $\\mathbf{T}$. We show that, by refining the index set to the level product $\\otimes\\mathbf{S}$ of a vector strong subtree $\\bfcs$ of $\\mathbf{S}$, such families of events become highly correlated. An analogue of Lebesgue's density Theorem is also established which can be considered as the \"probabilistic\" version of the density Halpern--L\\\"{a}uchli Theorem.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-06-13T19:33:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "measurable events",
      "level product",
      "vector homogeneous tree",
      "finite sequence",
      "lebesgues density theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.4988D"
    }
  }
}