{
  "id": "1911.00434",
  "version": "v1",
  "published": "2019-11-01T15:53:25.000Z",
  "updated": "2019-11-01T15:53:25.000Z",
  "title": "Consistency of $\\neg AC^{3}$ + $`χ(E_{G_{1}})=3$, $χ(E_{G_{2}})\\geqω\\impliesχ(E_{G_{1}\\times G_{2}})=3$' and relative consistency via strongly compactness",
  "authors": [
    "Amitayu Banerjee",
    "Zalán Gyenis"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove Andr\\'as Hajnal's \\textbf{Theorem 2} of \\cite{Haj1985} in different ways and observe a {\\em permutation model} where the {\\em axiom of choice for 3 element sets} fails but the statement in \\textbf{Theorem 2} of \\cite{Haj1985} still holds for $k=3$. We also observe that the {\\em Dilworth's decomposition theorem for infinite p.o.sets of finite width} holds and a weaker form of {\\L}o\\'{s}'s lemma (p. 253 of \\cite{HoRu1998}) fails in the permutation model of \\textbf{Theorem 7} of \\cite{HT2018} due to Lorenz Halbeisen and Eleftherios Tachtsis. Secondly, we weaken the large cardinal assumption of the results from \\cite{AC2013} due to Arthur Apter and Brent Cody, from a supercompact cardinal to a strongly compact cardinal. Further, applying the {\\em appropriate automorphism technique} from \\cite{AH1991} we remove the additional assumption that {\\em `every strongly compact cardinal is a limit of measurable cardinals'} from \\textbf{corollary 2.32} of section 4, chapter 2 of \\cite{Dim2011} by Ioanna Dimitriou.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-01T15:53:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "strongly compactness",
      "relative consistency",
      "strongly compact cardinal",
      "permutation model",
      "dilworths decomposition theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}