### Ideals without ccc

by Balcerzak and Roslanowski and Shelah. [BRSh:512]

J Symbolic Logic, 1998

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).

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