Higher Independence

2019/09/25 by Fischer, Vera, Montoya, Diana Carolina · 1 citation
#03E17 #03E35 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1909.11623

Abstract

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

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