Van Douwen and many non Van Douwen families
Lukas Schembecker
TL;DR
The paper investigates the spectra of maximal eventually different (m.e.d.) families and their Van Douwen refinements, proving that the Van Douwen spectrum $spec(\mathfrak{a}_{v})$ is closed under singular limits and that forcing notions like ${\mathbb{E}}_{\mathcal{F}}(I)$ can realize large mad subfamilies while preserving connections to $spec(\mathfrak{a}_{e})$. Under CH, it shows that any ideal containing Fin can be realized as the associated ideal $\mathcal{I}_{0}(\mathcal{F})$ of a m.e.d. family, and that these associated ideals can be made Sacks-indestructible, yielding robustness under iterated Sacks forcing. These results yield corollaries such as the realization of every $\aleph_{1}$-generated ideal in the iterated Sacks-model and the existence of very non Van Douwen m.e.d. families. Collectively, the work clarifies the relationship between $\mathfrak{a}_{e}$ and $\mathfrak{a}_{v}$, provides powerful forcing constructions to tailor associated ideals, and demonstrates strong persistence of these structures under canonical set-theoretic iterations.
Abstract
We prove that the spectrum of Van Douwen families is closed under singular limits. For any maximal eventually different family Raghavan defined in an associated ideal which measures how far the family is from being Van Douwen. Under CH we prove that every ideal containing Fin is realized as the associated ideal of some maximal eventually different family. Finally, we construct maximal eventually different families with Sacks-indestructible associated ideals to prove that in the iterated Sacks-model every $\aleph_1$-generated ideal containing Fin is realized.