Van Douwen Families, Productivity and Ultrafilter Maximality

Abstract

We study idealized maximal eventually different families through Van Douwen maximality, productivity, and ultrafilter maximality. We show that Van Douwen 𝒥-MED families correspond to (𝒥×∅)-MAD families, and for coanalytic ideals containing the finite sets they are further reduced to infinite analytic 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 𝒰^*-maximality for some ultrafilter 𝒰 extending 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 Fin-MED family exists. Finally, for every Ramsey ultrafilter 𝒰 and 1≤α<ω₁, there is no analytic (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.

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BibTeX RIS

BibTeX

@misc{he2026douwen,
  title = {{Van Douwen Families, Productivity and Ultrafilter Maximality}},
  author = {Jialiang He and Jintao Luo and Hang Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02350},
  eprint = {2610.02350},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02350},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Van Douwen Families, Productivity and Ultrafilter Maximality
AU  - Jialiang He
AU  - Jintao Luo
AU  - Hang Zhang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02350
UR  - https://arxiv.org/abs/2610.02350
AB  - We study idealized maximal eventually different families through Van Douwen maximality, productivity, and ultrafilter maximality. We show that Van Douwen 𝒥-MED families correspond to (𝒥×∅)-MAD families, and for coanalytic ideals containing the finite sets they are further reduced to infinite analytic 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 𝒰^*-maximality for some ultrafilter 𝒰 extending 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 Fin-MED family exists. Finally, for every Ramsey ultrafilter 𝒰 and 1≤α<ω_1, there is no analytic (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.
ER  -

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