{
  "id": "1505.01057",
  "version": "v1",
  "published": "2015-05-05T15:59:43.000Z",
  "updated": "2015-05-05T15:59:43.000Z",
  "title": "The strength of the tree theorem for pairs in reverse mathematics",
  "authors": [
    "Ludovic Patey"
  ],
  "comment": "15 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "No natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA) and Ramsey's theorem for pairs (RT^2_2) in reverse mathematics. The tree theorem for pairs (TT^2_2) is however a good candidate. The tree theorem states that for every finite coloring over tuples of comparable nodes in the full binary tree, there is a monochromatic subtree isomorphic to the full tree. The principle TT^2_2 is known to lie between ACA and RT^2_2 over RCA, but its exact strength remains open. In this paper, we prove that RT^2_2 together with weak K\\\"onig's lemma (WKL) does not imply TT^2_2, thereby answering a question of Montalban. This separation is a case in point of the method of Lerman, Solomon and Towsner for designing a computability-theoretic property which discriminates between two statements in reverse mathematics. We therefore put the emphasis on the different steps leading to this separation in order to serve as a tutorial for separating principles in reverse mathematics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-05T15:59:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03B30",
      "03F35"
    ],
    "keywords": [
      "reverse mathematics",
      "exact strength remains open",
      "tree theorem states",
      "full binary tree",
      "monochromatic subtree isomorphic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150501057P"
    }
  }
}