$\mathfrak{d}=ω_1$ implies $\mathfrak{a}=ω_1$

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.