Selective independence

TL;DR

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Contribution

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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 $\Delta^1_3$ 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 $\mathfrak{i}<\mathfrak{a}$.