Strong independence and its spectrum

TL;DR

This paper explores the spectrum of sizes of maximal independent families in higher Baire spaces for uncountable cardinals, revealing independence results and consistency of various sizes relative to large cardinals.

Contribution

It introduces the concept of the spectrum of maximal independent families for uncountable cardinals and demonstrates independence and consistency results related to their sizes.

Findings

The spectrum's non-emptiness is undecidable in ZFC with large cardinals.

Consistent existence of families with sizes between  and 2^ 0 for uncountable 0.

The spectrum cannot be arbitrary, indicating structural constraints.

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 $\bigcap\mathcal{A}_0\backslash\bigcup\mathcal{A}_1 \not= \emptyset$, $\mathcal{A}$ 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 $\mathfrak{sp}_{\mathfrak i}(\mu,\kappa)\neq\emptyset$ 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.