Strong independence and its spectrum

For & mu;, k infinite, say A C [k]& kappa; is a (& mu;, k)-maximal independent family if whenever A0 and A1 are pairwise disjoint non-empty in [A]<& mu; then ii A0\ U A1 =⠄ 0, A is maximal under inclusion among families with this property, and moreover all such Boolean combinations have size k. We denote by spi(& mu;, k) the set of all cardinalities of such families, and if non-empty, we let i & mu;(k) be its minimal element. Thus, i & mu;(k) (if defined) is a natural higher analogue of the independence number on w for the higher Baire spaces. In this paper, we study spi(& mu;, k) for & mu;, k uncountable. Among others, we show that: (1) The property spi(& mu;, k) =⠄0 cannot be decided on the basis of ZFC plus large cardinals. (2) Relative to a measurable, it is consistent that: (a) (Dk>w) i & kappa;(k) < 2 & kappa;. (b) (Dk>w) k+ < i & omega;1 (k) < 2 & kappa;. To the best knowledge of the authors, (2b) is the first example of a (& mu;, k)-maximal independent family of size strictly between k+ and 2 & kappa;, for uncountable k. (3) spi(& mu;, k) cannot be quite arbitrary. & COPY; 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).