Strong independence and its spectrum

Abstract

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