{
  "id": "2311.06872",
  "version": "v1",
  "published": "2023-11-12T15:13:34.000Z",
  "updated": "2023-11-12T15:13:34.000Z",
  "title": "Ramsey theorem for trees with successor operation",
  "authors": [
    "Martin Balko",
    "David Chodounský",
    "Natasha Dobrinen",
    "Jan Hubička",
    "Matěj Konečný",
    "Jaroslav Nešetřil",
    "Andy Zucker"
  ],
  "comment": "37 pages, 9 figures",
  "categories": [
    "math.CO",
    "cs.DM",
    "math.LO"
  ],
  "abstract": "We prove a general Ramsey theorem for trees with a successor operation. This theorem is a common generalization of the Carlson-Simpson Theorem and the Milliken Tree Theorem for regularly branching trees. Our theorem has a number of applications both in finite and infinite combinatorics. For example, we give a short proof of the unrestricted Ne\\v{s}et\\v{r}il-R\\\"odl theorem, and we recover the Graham-Rothschild theorem. Our original motivation came from the study of big Ramsey degrees - various trees used in the study can be viewed as trees with a successor operation. To illustrate this, we give a non-forcing proof of a theorem of Zucker on big Ramsey degrees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-12T15:13:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10",
      "05C05",
      "05C65",
      "05C55",
      "G.2.2",
      "F.4.1"
    ],
    "keywords": [
      "successor operation",
      "big ramsey degrees",
      "general ramsey theorem",
      "milliken tree theorem",
      "original motivation came"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}