Selective independence

Abstract

Let $\mathfrak{i}$ denote the minimal cardinality of a maximal independent family and let ${\mathfrak{a}}_T$ denote the minimal cardinality of a maximal family of pairwise almost disjoint subtrees of $2^{<\omega }$. Using a countable support iteration of proper, ${}^{\omega }\omega$-bounding posets of length ${\omega }_2$ over a model of CH, we show that consistently $\mathfrak{i}<{\mathfrak{a}}_T$. Moreover, we show that the inequality can be witnessed by a co-analytic maximal independent family of size ${\aleph }_1$ in the presence of a ${\Delta }_3^1$ definable well-order of the reals. The main result of the paper can be viewed as a partial answer towards the well-known open problem of the consistency of $\mathfrak{i}<\mathfrak{a}$.

8 pages