{
  "id": "2206.14936",
  "version": "v1",
  "published": "2022-06-29T22:42:25.000Z",
  "updated": "2022-06-29T22:42:25.000Z",
  "title": "Combinatorial properties of MAD families",
  "authors": [
    "Jörg Brendle",
    "Osvaldo Guzmán",
    "Michael Hrušák",
    "Dilip Raghavan"
  ],
  "comment": "43 pages. Submitted. arXiv admin note: text overlap with arXiv:1810.09680",
  "categories": [
    "math.LO",
    "math.GN"
  ],
  "abstract": "We study some strong combinatorial properties of $\\textsf{MAD}$ families. An ideal $\\mathcal{I}$ is Shelah-Stepr\\={a}ns if for every set $X\\subseteq{\\left[ \\omega\\right]}^{<\\omega}$ there is an element of $\\mathcal{I}$ that either intersects every set in $X$ or contains infinitely many members of it. We prove that a Borel ideal is Shelah-Stepr\\={a}ns if and only if it is Kat\\v{e}tov above the ideal $\\textsf{fin}\\times\\textsf{fin}$. We prove that Shelah-Stepr\\={a}ns $\\textsf{MAD}$ families have strong indestructibility properties (in particular, they are both Cohen and random indestructible). We also consider some other strong combinatorial properties of $\\textsf{MAD}$ families. Finally, it is proved that it is consistent to have $\\mathrm{non}(\\mathcal{M}) = {\\aleph}_{1}$ and no Shelah-Stepr\\={a}ns families of size ${\\aleph}_{1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-29T22:42:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mad families",
      "strong combinatorial properties",
      "strong indestructibility properties",
      "borel ideal",
      "consistent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}