{
  "id": "2405.18431",
  "version": "v1",
  "published": "2024-05-28T17:59:47.000Z",
  "updated": "2024-05-28T17:59:47.000Z",
  "title": "A Ramsey theorem for the reals",
  "authors": [
    "Tanmay Inamdar"
  ],
  "comment": "Preliminary version",
  "categories": [
    "math.LO",
    "math.CO",
    "math.GN"
  ],
  "abstract": "We prove that for every colouring of pairs of reals with finitely-many colours, there is a set homeomorphic to the rationals which takes no more than two colours. This was conjectured by Galvin in 1970, and a colouring of Sierpi{\\'n}ski from 1933 witnesses that the number of colours cannot be reduced to one. Previously in 1985 Shelah had shown that a stronger statement is consistent with a forcing construction assuming the existence of large cardinals. Then in 2018 Raghavan and Todor\\v{c}evi\\'c had proved it assuming the existence of large cardinals. We prove it in $ZFC$. In fact Raghavan and Todor\\v{c}evi\\'c proved, assuming more large cardinals, a similar result for a large class of topological spaces. We prove this also, again in $ZFC$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-28T17:59:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E02",
      "03E04",
      "03E55",
      "05D10",
      "05C55"
    ],
    "keywords": [
      "ramsey theorem",
      "large cardinals",
      "fact raghavan",
      "set homeomorphic",
      "large class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}