{
  "id": "1905.08425",
  "version": "v1",
  "published": "2019-05-21T03:29:46.000Z",
  "updated": "2019-05-21T03:29:46.000Z",
  "title": "The weakness of the pigeonhole principle under hyperarithmetical reductions",
  "authors": [
    "Benoit Monin",
    "Ludovic Patey"
  ],
  "comment": "32 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "The infinite pigeonhole principle for 2-partitions ($\\mathsf{RT}^1_2$) asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that $\\mathsf{RT}^1_2$ admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every $\\Delta^0_n$ set, of an infinite low${}_n$ subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak, Jockusch and Slaman.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-21T03:29:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperarithmetical reductions",
      "infinite pigeonhole principle",
      "admits strong cone avoidance",
      "infinite subset",
      "complement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}