($\Beta \Delta \mathca{I}$,$\mathca{I}$)-saturated sets and Hamel basis

Abstract

Let $\mathca{I}$ be a proper $\sigma$-ideal of subsets of the real line.

In a $\sigma$-field of Borel sets modulo sets from the $\sigma$-ideal $\mathca{I}$ we introduce an analogue of the saturated non-measurability
considered by I. Halperin.

Properties of ($\Beta \Delta \mathca{I}$,$\mathca{I}$)-saturated sets are investigated.
M. Kuczma considered a problem how small or large a Hamel basis can be.

We try tostudy this problem in the context of sets from I.