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 $\mathcal{J}$-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.