Strong independence and its spectrum

Abstract

For μ,κ infinite, say A⊆[κ]κ is a (μ,κ)-maximal independent family if whenever A0 and A1 are pairwise disjoint non-empty in [A]<μ then ⋂A0﹨⋃A1≠∅, A is maximal under inclusion among families with this property, and moreover all such Boolean combinations have size κ. We denote by spi(μ,κ) the set of all cardinalities of such families, and if non-empty, we let iμ(κ) be its minimal element. Thus, iμ(κ) (if defined) is a natural higher analogue of the independence number on ω for the higher Baire spaces. In this paper, we study spi(μ,κ) for μ,κ uncountable. Among others, we show that:(1)The property spi(μ,κ)≠∅ cannot be decided on the basis of ZFC plus large cardinals.(2)Relative to a measurable, it is consistent that:(a)(∃κ>ω)iκ(κ)<2κ.(b)(∃κ>ω)κ+<iω1(κ)<2κ. To the best knowledge of the authors, (2b) is the first example of a (μ,κ)-maximal independent family of size strictly between κ+ and 2κ, for uncountable κ.(3)spi(μ,κ) cannot be quite arbitrary.