(B∆I,I)-saturated sets and Hamel basis

Let I be a proper σ-ideal of subsets of the real line. In a σ-field of Borel sets modulo sets from the σ-ideal I we introduce an analogue of the saturated non-measurability considered by Halperin. Properties of (B∆I,I)-saturated sets are investigated.

M. Kuczma considered a problem how small or large a Hamel basis can be. We try to study this problem in the context of sets from I.