{
  "id": "1912.03327",
  "version": "v1",
  "published": "2019-12-06T19:38:53.000Z",
  "updated": "2019-12-06T19:38:53.000Z",
  "title": "Telgarsky's conjecture may fail",
  "authors": [
    "Will Brian",
    "Alan Dow",
    "David Milovich",
    "Lynne Yengulalp"
  ],
  "categories": [
    "math.LO",
    "math.GN"
  ],
  "abstract": "Telg\\'arsky's conjecture states that for each $k \\in \\mathbb N$, there is a topological space $X_k$ such that in the Banach-Mazur game on $X_k$, the player {\\scriptsize NONEMPTY} has a winning $(k+1)$-tactic but no winning $k$-tactic. We prove that this statement is consistently false. More specifically, we prove, assuming $\\mathsf{GCH}+\\square$, that if {\\scriptsize NONEMPTY} has a winning strategy for the Banach-Mazur game on a $T_3$ space $X$, then she has a winning $2$-tactic. The proof uses a coding argument due to Galvin, whereby if $X$ has a $\\pi$-base with certain nice properties, then {\\scriptsize NONEMPTY} is able to encode, in each consecutive pair of her opponent's moves, all essential information about the play of the game before the current move. Our proof shows that under $\\mathsf{GCH}+\\square$, every $T_3$ space has a sufficiently nice $\\pi$-base that enables this coding strategy. Translated into the language of partially ordered sets, what we really show is that $\\mathsf{GCH}+\\square$ implies the following statement, which is equivalent to the existence of the \"nice'' $\\pi$-bases mentioned above: \\emph{Every separative poset $\\mathbb P$ with the $\\kappa$-cc contains a dense sub-poset $\\mathbb D$ such that $|\\{ q \\in \\mathbb D \\,:\\, p \\text{ extends } q \\}| < \\kappa$ for every $p \\in \\mathbb P$.} We prove that this statement is independent of $\\mathsf{ZFC}$: while it holds under $\\mathsf{GCH}+\\square$, it is false even for ccc posets if $\\mathfrak{b} > \\aleph_1$. We also show that if $|\\mathbb P| < \\aleph_\\omega$, then \\axiom-for-$\\mathbb P$ is a consequence of $\\mathsf{GCH}$ holding below $|\\mathbb P|$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-06T19:38:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "banach-mazur game",
      "telgarskys conjecture states",
      "nice properties",
      "ccc posets",
      "opponents moves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}