  Let I be an ideal of subsets of a Polish space X, containing all singletons
and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider
the following conditions (B), (M) and (D). Condition (B) states that there is a
disjoint family F subseteq P(X) of size c, consisting of Borel sets which are
not in I. Condition (M) states that there is a function f:X-> X with
f^{-1}[{x}] notin I for each x in X. Provided that X is a group and I is
invariant, condition (D) states that there exist a Borel set B notin I and a
perfect set P subseteq X for which the family {B+x: x in P} is disjoint. The
aim of the paper is to study whether the reverse implications in the chain (D)
=> (M) => (B) => not-ccc can hold. We build a sigma-ideal on the Cantor group
witnessing''(M) and not (D)'' (Section 2). A modified version of that
sigma-ideal contains the whole space (Section 3). Some consistency results
deriving (M) from (B) for''nicely'' defined ideals are established (Section 4).
We show that both ccc and (M) can fail (Theorems 1.3 and 4.2). Finally, some
sharp versions of (M) for invariant ideals on Polish groups are investigated
(Section 5).
