{
  "abstract": "We study idealized maximal eventually different families through Van Douwen maximality, productivity, and ultrafilter maximality. We show that Van Douwen $\\mathcal{J}$-MED families correspond to $(\\mathcal{J}\\times\\emptyset)$-MAD families, and for coanalytic ideals containing the finite sets they are further reduced to infinite analytic $\\mathcal{J}$-MAD families. This yields nonexistence results for analytic Van Douwen families for several standard ideals, together with a characterization of the finite case. We recall the definition of finite productivity and introduce finite-section and $ω$-centered productivity. Finite-section productivity is equivalent to $\\mathcal{U}^*$-maximality for some ultrafilter $\\mathcal{U}$ extending $\\mathcal{J}^*$, and these notions admit natural Stone-space characterizations. We construct Borel finitely productive families for uniformly weakly Ramsey ideals and Borel $ω$-centered productive families for uniformly weakly $P^+$ ideals, while no analytic finite-section productive $\\mathrm{Fin}$-MED family exists. Finally, for every Ramsey ultrafilter $\\mathcal{U}$ and $1\\leqα<ω_1$, there is no analytic $(\\mathcal{U}^α)^*$-MED family, while under $V=L$, there is a $ \\boldsymbolΣ^1_2 $ Ramsey ultrafilter $ \\mathcal{U} $ where there is a coanalytic $ \\mathcal{U}^* $-MED family.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02350",
  "author_total": 3,
  "authors": [
    {
      "name": "Jialiang He"
    },
    {
      "name": "Jintao Luo"
    },
    {
      "name": "Hang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3mash22pcqnqc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02350",
  "fulltext_url": null,
  "id": "3mash22pcqnqc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02350",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.02350",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3mash22pcqnqc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02350",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02350"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Van Douwen Families, Productivity and Ultrafilter Maximality",
  "title_text": "Van Douwen Families, Productivity and Ultrafilter Maximality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/3mash22pcqnqc",
  "venue": "arXiv",
  "versions": []
}