Completely nonmeasurable unions

Assume that no cardinal κ < 2 ^ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal $ \mathbb{I} $ \mathbb{I} of X contains uncountably many pairwise disjoint subfamilies $ \mathbb{I} $ \mathbb{I} -Bernstein unions ∪ $ \mathbb{I} $ \mathbb{I} -Bernstein if A and X \ A meet each Borel $ \mathbb{I} $ \mathbb{I} -positive subset B ⊆ X). This result is a generalization of the Four Poles Theorem (see 1]) and results from 2] and 4].