{
  "id": "1803.09771",
  "version": "v2",
  "published": "2018-03-26T18:07:53.000Z",
  "updated": "2018-07-17T20:38:50.000Z",
  "title": "Pigeons do not jump high",
  "authors": [
    "Benoit Monin",
    "Ludovic Patey"
  ],
  "comment": "20 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "The infinite pigeonhole principle for 2-partitions asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we develop a new notion of forcing enabling a fine analysis of the computability-theoretic features of the pigeonhole principle. We deduce various consequences, such as the existence, for every set $A$, of an infinite subset of it or its complement of non-high degree. We also prove that every $\\Delta^0_3$ set has an infinite low${}_3$ solution and give a simpler proof of Liu's theorem that every set has an infinite subset in it or its complement of non-PA degree.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-07-17T20:38:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "jump high",
      "infinite subset",
      "infinite pigeonhole principle",
      "complement",
      "non-pa degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}