{
  "id": "1405.0460",
  "version": "v1",
  "published": "2014-05-02T18:00:32.000Z",
  "updated": "2014-05-02T18:00:32.000Z",
  "title": "Distinguishing subgroups of the rationals by their Ramsey properties",
  "authors": [
    "Ben Barber",
    "Neil Hindman",
    "Imre Leader",
    "Dona Strauss"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A system of linear equations with integer coefficients is partition regular over a subset S of the reals if, whenever S\\{0} is finitely coloured, there is a solution to the system contained in one colour class. It has been known for some time that there is an infinite system of linear equations that is partition regular over R but not over Q, and it was recently shown (answering a long-standing open question) that one can also distinguish Q from Z in this way. Our aim is to show that the transition from Z to Q is not sharp: there is an infinite chain of subgroups of Q, each of which has a system that is partition regular over it but not over its predecessors. We actually prove something stronger: our main result is that if R and S are subrings of Q with R not contained in S, then there is a system that is partition regular over R but not over S. This implies, for example, that the chain above may be taken to be uncountable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-02T18:00:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10"
    ],
    "keywords": [
      "partition regular",
      "ramsey properties",
      "distinguishing subgroups",
      "linear equations",
      "infinite chain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0460B"
    }
  }
}