Strong independence and its spectrum

For $μ, κ$ infinite, say $\mathcal{A}\subseteq [κ]^κ$ is a $(μ,κ)$-maximal independent family if whenever $\mathcal{A}_0$ and $\mathcal{A}_1$ are pairwise disjoint non-empty in $[\mathcal{A}]^{<μ}$ then $\bigcap\mathcal{A}_0\backslash\bigcup\mathcal{A}_1 \not= \emptyset$, $\mathcal{A}$ is maximal under inclusion among families with this property, and moreover all such Booelan combinations have size $κ$. We denote by $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ the set of all cardinalities of such families, and if non-empty, we let $\mathfrak{i}_μ(κ)$ be its minimal element. Thus, $\mathfrak{i}_μ(κ)$ (if defined) is a natural higher analogue of the independence number on $ω$ for the higher Baire spaces. In this paper, we study $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ for $μ,κ$ uncountable. Among others, we show that: (1) The property $\mathfrak{sp}_{\mathfrak i}(μ,κ)\neq\emptyset$ cannot be decided on the basis of ZFC plus large cardinals. (2) Relative to a measurable, it is consistent that: (a) $(\exists κ{>}ω) \, \mathfrak{i}_κ(κ)<2^κ$; (b) $(\exists κ{>}ω)\,κ^+<\mathfrak{i}_{ω_1}(κ)<2^κ$. To the best knowledge of the authors, this is the first example of a $(μ,κ)$-maximal independent family of size strictly between $κ^+$ and $2^κ$, for uncountable $κ$. (3) $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ cannot be quite arbitrary.

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