On Borel sets belonging to every invariant ccc $\sigma$-ideal on $2^{\mathbb {N}}$

Abstract:

Let $I_{ccc}$ be the $\sigma$-ideal of subsets of the Cantor group $2^{\mathbb {N}}$ generated by Borel sets which belong to every translation-invariant $\sigma$-ideal on $2^{\mathbb {N}}$ satisfying the countable chain condition (ccc). We prove that $I_{ccc}$ strongly violates ccc. This generalizes a theorem of Balcerzak-Rosłanowski-Shelah stating the same for the $\sigma$-ideal on $2^{\mathbb {N}}$ generated by Borel sets $B\subseteq 2^{\mathbb {N}}$ which have perfectly many pairwise disjoint translates. We show that the last condition does not follow from $B\in I_{ccc}$ even if $B$ is assumed to be compact. Various other conditions which for a Borel set $B$ imply that $B\in I_{ccc}$ are also studied. As a consequence we prove in particular that:

If $A_n$ are Borel sets, $n\in \mathbb N$, and $2^{\mathbb {N}}=\bigcup _n A_n$, then there is $n_0$ such that every perfect set $P\subseteq 2^{\mathbb {N}}$ has a perfect subset $Q$, a translate of which is contained in $A_{n_0}$.

CH is equivalent to the statement that $2^{\mathbb {N}}$ can be partitioned into $\aleph _1$ many disjoint translates of a closed set.