{
  "id": "2506.17943",
  "version": "v1",
  "published": "2025-06-22T08:36:00.000Z",
  "updated": "2025-06-22T08:36:00.000Z",
  "title": "Finite Combinatorics and Fragments of Arithmetic",
  "authors": [
    "Wei Wang"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "In fragments of first order arithmetic, definable maps on finite domains could behave very differently from finite maps. Here combinatorial properties of $\\Sigma_{n+1}$-definable maps on finite domains are compared in the absence of $B\\Sigma_{n+1}$. It is shown that $\\mathrm{GPHP}(\\Sigma_{n+1})$ (the $\\Sigma_{n+1}$-instance of Kaye's General Pigeonhole Principle) lies strictly between $\\mathrm{CARD}(\\Sigma_{n+1})$ and $\\mathrm{WPHP}(\\Sigma_{n+1})$ (Weak Pigeonhole Principle for $\\Sigma_{n+1}$-maps), and also that $\\mathrm{FRT}(\\Sigma_{n+1})$ (Finite Ramsey's Theorem for $\\Sigma_{n+1}$-maps) does not imply $\\mathrm{WPHP}(\\Sigma_{n+1})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-22T08:36:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03F30",
      "03C20",
      "03H15"
    ],
    "keywords": [
      "finite combinatorics",
      "finite domains",
      "kayes general pigeonhole principle",
      "definable maps",
      "first order arithmetic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}