Higher Independence

TL;DR

This paper explores higher analogues of the classical independence number for uncountable cardinals, establishing relations with other cardinal characteristics and constructing models demonstrating their possible values.

Contribution

It introduces the concept of $i()$ for uncountable $$, establishes ZFC relations with other characteristics, and constructs models showing specific equalities and inequalities.

Findings

Established ZFC relations between $i()$ and classical cardinal characteristics.

Constructed models where $^+=()=()$ and $2^$ can be larger.

Discussed open problems and challenges in extending higher independence concepts.

Abstract

We study higher analogues of the classical independence number on $\omega$. For $\kappa$ regular uncountable, we denote by $i(\kappa)$ the minimal size of a maximal $\kappa$-independent family. We establish ZFC relations between $i(\kappa)$ and the standard higher analogues of some of the classical cardinal characteristics, e.g. $\mathfrak{r}(\kappa)\leq\mathfrak{i}(\kappa)$ and $\mathfrak{d}(\kappa)\leq\mathfrak{i}(\kappa)$. For $\kappa$ measurable, assuming that $2^\kappa=\kappa^+$ we construct a maximal $\kappa$-independent family which remains maximal after the $\kappa$-support product of $\lambda$ many copies of $\kappa$-Sacks forcing. Thus, we show the consistency of $\kappa^+=\mathfrak{d}(\kappa)=\mathfrak{i}(\kappa)<2^\kappa$. We conclude the paper with interesting open questions and discuss difficulties regarding other natural approaches to higher independence.