{
  "abstract": "We prove in ZFC that $\\mathfrak d=ω_1$ implies $\\mathfrak a=ω_1$, settling a question of Roitman from the 1970s. Here $\\mathfrak d$ is the least size of a family that eventually dominates every function in $ω^ω$, and $\\mathfrak a$ is the least size of an infinite maximal almost disjoint (MAD) family of infinite subsets of $ω$. Consequently, $\\mathfrak a\\leq\\mathfrak d$ whenever the continuum $\\mathfrak c=2^{\\aleph_0}$ is at most $ω_2$. This answers negatively Shelah's question whether $\\mathfrak d<\\mathfrak a$ is consistent with $\\mathfrak c=ω_2$, and shows that his model of $\\mathfrak d=ω_2<\\mathfrak a=\\mathfrak c=ω_3$ has the least possible values of $\\mathfrak d$, $\\mathfrak a$ and $\\mathfrak c$ among models of $\\mathfrak d<\\mathfrak a$. The proof also gives a preservation theorem: from any dominating family of size $ω_1$, we construct a MAD family of size $ω_1$ that remains maximal in every outer model in which the given family is still dominating. Thus, under $\\mathfrak d=ω_1$, some MAD family is indestructible by every $ω^ω$-bounding forcing; it can moreover be chosen almost strongly separable, hence Cohen-indestructible. These results give negative answers, under $\\mathfrak d=ω_1$, to two questions of Hrušák and to one of Brendle, Guzmán, Hrušák and Raghavan.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00639",
  "author_total": 1,
  "authors": [
    {
      "name": "José de Jesús Pelayo Gómez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/35whe6mfes7t9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00639",
  "fulltext_url": null,
  "id": "35whe6mfes7t9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00639",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7216347922",
  "pdf_url": "https://arxiv.org/pdf/2610.00639",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/35whe6mfes7t9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00639",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00639"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "$\\mathfrak{d}=ω_1$ implies $\\mathfrak{a}=ω_1$",
  "title_text": "d=ω_1 implies a=ω_1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/35whe6mfes7t9",
  "venue": "arXiv",
  "versions": []
}